im-pmf.weebly.comim-pmf.weebly.com/uploads/5/8/9/8/58988609/01-bilten_ii-2013-1.pdf · Matematiqki Bilten ISSN 0351-336X (print) 37(LXIII) No. 2 ISSN 1857-9914 (online) 2013(5-82)

Matematiqki Bilten ISSN 0351-336X (print)37(LXIII) No. 2 ISSN 1857-9914 (online)2013(5-82) UDC:510.38:510.426:515.122Skopje, Makedonija

SKAND THEORY AND ITS APPLICATIONS.(A NEW LOOK AT NON-WELL-FOUNDED SETS)

JU. T. LISICA

Abstract. A new mathematical object called a skand is introduced,

which turns out in general to be a non-well-founded set. Skands offinite lengths are ordinary well-founded sets, and skands of very long

length (like the hyper-skand of all ordinals) are hyper-classes.Self-similar skands are also considered, and they clarify the reflex-

ivity of sets, i.e., the meaning of the relation X ∈ X ; in particular,

self-similar skands considered as non-well-founded sets are always re-flexive, but not vice versa. The existence of self-similar skands shows

at once that all the well-known set-theoretical paradoxes are not para-doxes at all, and hence are not necessarily fatal for any set theory. E.g.,

the inconsistency of Russell’s “set” R = {X | X �∈ X} is proved here notwith the help of Russell’s paradox (as it is traditionally given, which is

incorrect), but via a simple method of the maximality (universality) of

R which goes back to Cantor and is also applied to other set-theoreticalparadoxes.

Generalized skands are also defined and a new look at the general-ized skand-class of all ordinals is demonstrated. In particular, the last

(class) ordinal called the eschaton is defined.

The next application of skand theory is a description of all epsilon-numbers in the sense of Cantor. Another application is a generalized

theory of one-dimensional continua of arbitrary powers and the con-struction of generalized real numbers as a non-Archimedean straight

line of arbitrary power, and the introduction of the absolute continuumand the absolute straight line as the hyper-classes nearest to the class

of sets.

2010 Mathematics Subject Classification. Primary 03E65, 03C62, 54G99.Key words and phrases. Skand, Russell’s paradox, non-well-founded sets, reflexive

sets, eschaton, epsilon-numbers, generalized rationals and reals, straight line of a largepower, generalized continuum.

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6 JU. T. LISICA

0. EpigraphsA parson had a hound-dog,One he loved a lot.It ate a piece of mutton,For which he had it shot.He buried the hound,Then wrote on its mound,ThatA parson had a hound-dog, etc.(ad infinitum and ad imum).[Free translation of a Russian children’s

ditty.]

Once a four-year-old son returned home from kindergarten,where he had been told that his father was a mathemati-

cian.When the son saw his father again, he asked him:“Is it true, Daddy, that you are a mathematician?”“Yes, sonny, it is,” was the answer.“Well,” responded the son, “can you count to the lastnumber?”“Ummm...ummmm,” mumbled the father, stumped.[Dialogue with a child which is in fact a problem ofmathematical eschatology:“What is the ’εσχατoν or ad imum?”]

“A soul is only a skand, i.e., an accidental aggregation ofbeing”.

[From Buddhist doctrine.]

“The content of a concept diminishes as its extensionincreases; if its extension becomes all-embracing, itscontent must vanish altogether”.[Gottlob Frege, “The Foundations of Arithmetic”.]

0. IntroductionWe are going to clarify the notion of reflexivity in Set Theory, i.e., the

meaning of a binary relation X ∈ Y in the case when Y = X , that is X ∈ X .As to Russell himself, the relation X ∈ X “must be always meaningless”[58], p. 81, since he was seriously frightened by this paradox, which waslater on named after him: “Thus X ∈ X was held to be meaningless,

SKAND THEORY AND ITS APPLICATIONS 7

because ∈ requires that the relatum should be a class composed of objectswhich are of the type of the referent” ([54], Chap. X, p. 107). Moreover,he went further and concluded that X �∈ X “must be always meaningless”,too. ([58], p. 81). He wrote: “If α is a class, the statement ‘α is not amember of α’ is always meaningless, and there is therefore no sense in thephrase ‘the class of those classes which are not members of themselves’ ”([58], p. 66). In particular, Russell wrote: “A class consisting of onlyone member must not be identical with that one member”. And he addedimmediately “X = {X} must be absolutely meaningless, not simply false”([58], p. 81). In this paper we shall see when Russell was right and whenwas he not.

In particular, we shall show that in many well-known formal systems(set theories) Russell’s paradox is not a paradox at all, and hence the proofof the Proposition that Russell’s collection R = {X |X /∈ X} is a properclass, not a set, via Russell’s paradox, is not correct (it was a logical mistakein the propositional calculus, at least within set theories with the axiom ofregularity). We prove this Proposition by the following

Maximality Principle. If there exists a maximal (universal) collectionX (sets, classes, hyper-classes), given by some property, predicate, etc.,then any assertion which implies the existence of a new element x with thesame property and x /∈ X is false.

2. Premises, notations, some history, and purpose of paperAt the beginning we start out within a von Neumann-Bernays-Godel-

type set theory (NBG for short), i.e., the theory of first-order logic withequality (i.e., with respect to the given definition of equality A2

1(X, Y )) witha syntactic or proof-theoretic side and a semantic or model-theoretic side(see [10], p. 7) with only the predicate letter A2

2(X, Y ), which is the binaryrelation X ∈ Y , for short. The proper axioms of NBG consist of generalaxioms, class-formation axioms and set-formation axioms together with theaxiom of choice AC and the axiom of foundation FA (see, e.g., in [41], p.225-286; we have changed only notation: NBG+(AC)+(Reg) on NBG).The basic set theory NBG can be considered with individuals (e.g., [41], p.297-304) called sometimes atoms or urelements, i.e., mathematical objectswhich are neither sets nor classes and which have no members, or NBGcan be considered without them; it does not matter. In the latter case theonly individual is just the empty set {} = ∅ and Set Theory in this case isoften called the theory of “pure” sets.

To be more precise and definite we denote by NBG[U ] the set theorywith of individuals (atoms, urelements), the class of all sets of NBG[U ] byV[U ], and the set or class of individuals (atoms, urelements) by U ; V is

8 JU. T. LISICA

the class of all sets in the theory NBG of “pure” sets, i.e., when U = ∅. InNBG[U ] because of FA the class V[U ] of all sets turns out to be the classWF of all well-founded sets. The universal class of elements U[U ] is theunion V[U ] ∪ U of V[U ] and U . Note that V[U ] ∩ U = ∅.

We also denote the class of all ordinals by On and the class of all car-dinals by Card.

Then we shall consider NBG[U ]−, i.e., NBG[U ] without the axiom offoundation FA, and instead of the axiom of choice we use the axiom N ofvon Neumann, V[U ]− ≈ On; i.e., these classes are bijective, which in theabsence of foundation is stronger than choice.

There is an important distinction in NBG[U ] as well as in NBG[U ]− SetTheories between three kinds of objects: individuals, sets (usually called“small” classes), and classes (usually called “large” classes or more oftenproper classes). Individuals do not contain any elements but can be ele-ments of non-empty sets and classes; non-empty sets can contain individualsand sets as elements but do not contain classes as elements, and sets can beelements of non-empty sets and clases; classes can obtain individuals andsets as elements but can not be elements of individuals, sets or classes. Theonly indivinual ∅ is called a set; others are not sets or classes.

There are two ways to distinguish sets and classes (i.e., proper classes)in addition to axioms for sets. The first of them is the following: a subclassX ⊂ U[U ] of U[U ] is a set if and only if there exists a one-element object(singleton) {X} ∈ V[U ]; the second one: a subclass X ⊂ U[U ] of U[U ] isa set if and only if there is no bijection X on U[U ]. Otherwise, X ⊆ U[U ]is not a set but a subclass (i.e., proper subclass) of U[U ] (further, in short,a class). Moreover, all subclasses (which are not sets) of U[U ] are bijectiveto each other. Notice also that when we deal with classes (i.e., properclasses) we speak in the language of their elements but not of them aswholes or as units. In other words, sets are arguments (elements of V[U ]),and classes are extensions of some predicates. In formal systems one usesthe following notations: Cls(X) as “X is a class” , M(X) as “X is a set” ,i.e., (∃Y )(X ∈ Y ), Pr(X) as “X is a proper class, i.e., Cls(X) ∧ ¬M(X),Ur(X) as “X is an urelement” and El(X) as “X is an elements” i.e.,El(X) = M(X) ∨ Ur(X). (See [41], Chap. 4, p. 297.) For simplicity weavoid here almost these notations.

Proposition 1. If theory NBG− is consistent, then NBG is also con-sistent.

Proof see in [40] or in [41], Chap. 4, 4. 86.


Moreover, a model of NBG is built by the following transfinite recursion:

Ψ(0) = ∅Ψ(α′) = PΨ(α)

lim(λ) =⇒ Ψ(λ) =⋃

β<λ

Ψ(β)

H′ =⋃

α∈On

Ψ(α).

(1)

Here and below α′ = α+1, α ∈ On, and P - the functor “set of all subsets”.Moreover, H′ determies an inner model of NBG− and the axiom of

foundation FA is equivalent to the statement that V = H′ (Ibidem).Proposition 2. Theory NBG[U ] is consistent if and only if NBG is

consistent.Proof see in [47] or in [41], Chap. 4, Proposition 4.50, p. 301-302.A model of NBG[U ] is built by the following transfinite recursion.If U is a set, then

Ξ(0) = UΞ(α′) = PΞ(α)

lim(λ) =⇒ Ξ(λ) =⋃

β<λ

Ξ(β)

H′[U ] =⋃

α∈On

Ξ(α).

(2)

Moreover, H′[U ] determines an inner model of NBG[U ]− and the axiomof foundation FA is equivalent to the statement that V[U ] = H′[U ].(Ibidem.)

If U is a class, then for each subset L ⊂ U and any ordinal γ ∈ On wedefine a set Ξγ

L be the following transfinite recursion:

ΞγL(0) = L

ΞγL(α′) = PΞγ

L(α), α′ < γ,lim(λ) =⇒ Ξγ

L(λ) =⋃

β<λ

ΞγL(β), λ < γ.

(3)

Let H[U ] be the class of all elements M such that for some set L andordinal γ, M is in the range of Ξγ

L. Then H[U ] determines a model ofNBG[U ] and the foundation axiom FA holds if and only if H[U ] = U[U ].Moreover, H[U ] determines an inner model of NBG[U ]−. (Ibidem.)

Recall that a set X is well-founded (or ordinary) if there is no infinite∈-descending chains, in other words, if every ∈-descending chain in X isfinite, i.e., for each x0 ∈ X , every ∈-descending chain starting with x0

can be at most the following one: x0 � x1 � x2 � ... � xn, where xn

is some individual or the empty set; otherwise it is non-well-founded or“extraordinary”, i.e., there exist infinite ∈-descending chains x0 � x1 �

10 JU. T. LISICA

x2 � ... � xn � .... The distinction between well-founded and non-well-founded sets was first articulated by Mirimanoff [43] and in his terminologythe distinction was between “ordinary” and “extraordinary” sets. Later on,von Neumann [50] proposed an axiom of regularity (“Restrictive Axiom”in [5] and “Axiom der Fundierung” in [60], i.e. the Foundation Axiom FA)which excluded Mirimanoff’s extraordinary sets, because according to theaxiom any “descending” sequence terminates, i.e., reaches its bottom or“foundation”. More precisely, the Axiom of Regularity in NBG[U ] isthe following:

(∀X)(X �= ∅) =⇒ (∃u)(u ∈ X ∧ ¬(∃v)(v ∈ X ∧ v ∈ u))) (4)

and together with the Axiom of Choice it is equivalent to the Axiom ofFoundation (see [41], Chap. 4, Proposition 4.44).

The restriction axiom as FA was very important, first of all, for com-pleteness of the extensionality axiom Ext, because in NBG[U ]− it is im-possible to prove in general that for different sets X and Y the two-elementset Z = {X, Y }, which always exists by the axiom of pairing, Z = X , orZ = Y , or Z is different from X and Y . Only by means of FA can oneprove that {X, Y } is different from X and Y . Secondly, FA avoids viciouscircle phenomena, i.e., there is no set X such that X � X1 � ... � Xn � X ;in particular, the reflexive sets X ∈ X , which appeared to be a source ofparadoxes (which was actually not true), e.g., Russell, with reference to H.Poincare wrote: “An analysis of the paradoxes to be avoided shows thatthey all result from a certain kind of vicious circle” [58], p. 39. And Russellhad formulated his famous “vicious-circle principle” as follows: “Whateverinvolves all of a collection must not be one of the collection” [58], p. 40.Thus, with the help of the Foundation Axiom, Set Theory has beensuccesfully developed and the “vicious-circle principle” has been satisfied.

Nevertheless, many real problems concern circular phenomena in logic(first of all, Russell’s paradox itself, because it concerns the relation R ∈ R,and therefore the non-well-founded set R; and also, e.g., the treatment ofLiar-like paradoxes, etc.); linguistics, computer science, graph theory, gametheory, streams, etc., all need circular models which lie beyond the universeof well-founded sets. At last, there is a lack of investigation of reflexive setsand Mirimanoff extraordinary sets which are in general many-valued. Evena real example of a set X which is an element of itself is absent in almostmonographs on Set Theory except those like “set of all sets”, e.g., [27],p. 5, or in Russell’s paradox R ∈ R in proving that R is not a set; theseexamples are fakes. Only later in monographs on “non-well-founded sets”real examples appeared.


From the early 20th century, many authors actually proposed their ownanti-foundation axioms (AFA for short) enriching and extending the Well-Founded Universe by the AFA-Universe, e.g., [25], [7], [50], [55], [26] andothers. All of them proposed theories of possibly non-well-founded setswhich are consistent, assuming that NBG− is consistent. Nevertheless,the four axiom systems mentioned are non-comparable, and each one differsfrom the others in the strengthening of the extensionality criterion for setequality.

In the present paper we introduce a new object called a skand (San-skrit: jump, skip). Skands are “definite and separate” objects (“bestimmtenwohlunterschiedenen Objekten” according to Cantor [15], p. 481) and canbe considered as elements of the class V[U ]− of all sets in NBG−, and theyalso enrich V[U ]; i.e., they can be well-founded or non-well-founded sets.Moreover, skands are essential extensions of some (not all) of Mirimanoff’sextraordinary sets.

The class of all skands generates a class (i.e., proper class) V[U ](1) whichis a subclass of the class V[U ]− by forming objects X whose elements are

ordinary sets or skands such that {X} exists. Clearly, V[U ]def= V[U ](0) ⊂

V[U ](1) ⊂ V[U ]−. Moreover, we successively continue such a process ofenrichment for each ordinal number α and obtain the following embed-dings: V[U ](0) ⊂ V[U ](1) ⊂ ... ⊂ V[U ](ν) ⊂ ... ⊂ V[U ]Ω ⊂ V[U ]−, whereV[U ]Ω =

⋃ν∈On

V[U ](α). We do not say that this cumulative hierarchy ex-

hausts V[U ]−, i.e., V[U ]Ω = V[U ]− but it makes V[U ]− more structural.�

2. The notion of a skandLet (α0, α)

def= {α′ ∈ On| α0 ≤ α′ < α} be a segment of On; if α0 = 0

and α ≥ 1, then we call (0, α) an initial segment of On.Definition 1. Consider a system of embedded curly braces or well-

ordered set of embedding pairs of curly braces {α0{α0+1...{α′...α′}...α0+1}α0},indexed by ordinals α′ ∈ (α0, α). We call it a trivial skand of lengthl = α − α0 and denote it by e(α0,α). It can be also called an emptyskand because for each α′ ∈ (α0, α) the set of all elements between twoneighboring opening braces {α′ {α′+1 or possibly between {α′ α′}, if α′ =α − 1, is empty. In other words, each α′-component eα′ of e(α0,α) isempty. By a non-trivial skand X(α0,α) of length l = α − α0 we under-stand a non-trivial system of embedded curly braces, indexed by ordi-nals α′ ∈ (α0, α); i.e., there is at least one index α′ ∈ (α0, α), and el-

ements x0, x1, x2, ..., xλ, ... of U[U ] such that the α′-component Xα′def=

12 JU. T. LISICA

{α′ x0, x1, x2, ..., xλ, ..., {α′+1 (or {α′ x0, x1, x2, ..., xλ, ... α′}, if α′ = α− 1) ofX(α0,α) is a set, i.e., {x0, x1, x2, ..., xλ, ...} ∈ V[U ] and denoted also as Xα.

For simplicity we shall omit indexes α′ of braces or elements in the caseswhen it is clear what they are, e.g., for {α′xα′

0 , xα′1 , xα′

2 , ..., xα′λα′, ..., {α′+1 or

possibly {α′xα′0 , xα′

1 , xα′2 , ..., xα′

λα′ , ...α′} in the case α′ = α − 1, we write

{xα′0 , xα′

1 , xα′2 , ..., xα′

λα′, ..., { and {xα′0 , xα′

1 , xα′2 , ..., xα′

λα′, ...} in the case α′ =α − 1, respectively; or even simplify to {x0, x1, x2, ..., xλ, ..., { and{x0, x1, x2, ..., xλ, ...}, respectively. All the more indexes α′ of braces areconditional, e.g., if X(0,α) is a skand, then Y(0,α) = {X(0,α)} whose compo-nents are Y0 = ∅, Y1 = X0, Y2 = X1, ..., Yn+1 = Xn, ... , and Yα′ = Xα′ ,for all ω ≤ α′ < α, is a skand, too.

For the purpose of interpreting skands as sets, we write a comma beforethe second open brace of non-trivial component Xα′ of X(α0,α), i.e.,{α′ x0, x1, x2, ..., xλ, ..., α′+1{ or simply {x0, x1, x2, ..., xλ, ..., {. It says thatbraces are not only syntactical but also semantic in the definition of skand.

Thus, a general form of an arbitrary skand is the following:

X(α0,α) = {xα00 , xα0

1 , ..., xα0λα0

, ..., {... {xα′0 , xα′

1 , ..., xα′λα′ , ..., {...}}...}, (5)

where components Xα′ = {xα′0 , xα′

1 , ..., xα′λα′ , ..., { are sets, i.e.,

{xα′0 , xα′

1 , ..., xα′λα′, ...} ∈ V[U ]

or empty, i.e., Xα′ = {{; if α is not a limit ordinal, then the last componentXα−1 is an ordinary set, i.e., {xα′

0 , xα′1 , ..., xα′

λα′, ...} ∈ V[U ] or empty {}.If all components Xα′ , α0 ≤ α′ < α, of a skand X(α0,α) are equal to

the same set, e.g., X = {x0, x1, ..., xλ, ...}, we shall denote this skand in ashorter way by X(α0,α)(X); in particular, when X = {γ}, i.e., a one-elementset X , we simplify the notation to X(α0,α)(γ). If α0 ≤ α1 < α2 ≤ α,then together with X(α0,α) we shall denote by X(α1,α2) the skand whoseα′-components Xα′ , α1 ≤ α′ < α2, are the same as those of X(α0,α), andX(α1,α2) is called a restriction of X(α0,α) on (α1, α2).

Definition 2. Two skands X(α0,α) and Y(β0,β) are called equal if thesegments (α0, α) and (β0, β) are isomorphic as well-ordered sets, i.e., their“similarity” is given by ϕ : (α0, α) → (β0, β), and the corresponding α′-and β′-componentsXα′ = {α′xα′

1 , xα′2 , ..., xα′

λ , ..., {α′+1 and Yβ′ = {β′yβ′1 , yβ′

2 , ..., yβ′λ , ..., {β′+1

as well as the last components Xα′ = {α′xα′1 , xα′

2 , ..., xα′λ , ...α′} and

Yβ′ = {β′yβ′1 , yβ′

2 , ..., yβ′λ , ...β′}, if α′ = α−1 and β′ = β−1, are equal as sets

in NBG[U ]; i.e., {xα′1 , xα′

2 , ..., xα′λ , ...} = {yβ′

1 , yβ′2 , ..., yβ′

λ , ...}, α0 ≤ α′ < αand β0 ≤ β′ < β, respectively, where β′ = ϕ(α′).


It is clear that the relation of equality is an equivalent relation.Remark 1. Actually, between each pair of braces {{ or {} of a skand

X(α0,α) there are different elements (sets or individuals), or their lack, whichform a set; notice that the order in which the members of sets are writtendoes not matter. In our notation we use Xα′ = {x0, x1, ..., xλ, ..., { or Xα′ ={x0, x1, ..., xλ, ...} only for simplicity of writing, since with the axiom ofchoice we can well-order any set X , λ < κ, for some ordinal κ, and obtainsuch notation. It is clear that the definition of a skand does not dependon orderings of elements of Xα′ , α0 ≤ α′ < α, which may be different,or notations, which may also be alternative. Thus, an arbitrary skand is“only an accidental aggregation” of well-founded sets or individuals eachof which figuratively “skips” into its own place, {{ or {}, in the system ofwell-ordered embedded curly braces; meanwhile there can be empty placesas well as a well-founded set, or an individual, and can “skip” into differentplaces and even be at all places at once; i.e., elements of each componentare always different and at the same time it may happen that some or evenall elements of different components may be equal.

Remark 2. Now we want to dispose of possible objections from theside of some logicians who might say that “braces are not objects of SetTheory and hence, e.g., a trivial skand e(α0,α) is not well-defined if α −α0 > ω”, or “an expression such as {...{{{∅}}}...} is not definite”, or thefollowing, which is thoroughly snobbish: “It would be better if set theoryteachers (and books on set theory) said at the outset that it is essential toview the universe of sets as a container intended to contain boxes intendedfor other boxes and one of them is intended to remain empty. Of coursethe setbrackets {...} suggest this view of sets, but this notion should beexplained to those students who meet it for the first time” [48], p. 534.

However we say that all “pure, well-founded sets” are systems of embed-ded curly braces; an expression {...{{{∅}}}...} is not definite, indeed, butanother expression {{{...{∅}...}}} of infinite pairs of braces is definite.

In any case the meaning of pairs of braces in the definition of skandis similar as in the definition of “pure well-founded sets” arising from theempty set ∅ = {}:

{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}, ...

and we use them only from a most possible convnience outward shape ofskand and an attempt to make it visualize.

Actually we can express a skand without any pairs of braces or bracketsusing only the binary relation ∈. For this purpose we recall the definitionof parametric family of sets [12]), resume, 2, 14. In our case it is a mapf(α0,α) : (α0, α) → V[U ], 0 ≤ α0 ≤ α, α0, α ∈ On, or more precisely, a

14 JU. T. LISICA

discrete union⊔

α′∈(α0,α)

Yα′ , where Yα′ is a singleton whose the only element

is f(α0,α)(α′), i.e., f(α0,α)(α′) ∈ Yα′ , α0 ≤ α′ < α.In our case (α0, α) is a set of parameters and elements of a parametric

sets are elements of⊔

α′∈(α0,α)

Yα′ and equal to f(α0,α)(α′) ∈ V[U ], α0 ≤ α′ <

α.

Now for an arbitrary skand X(α0,α) one can definitely associate a mapf(α0,α) : (α0, α) → V[U ] such that for each α′, α0 ≤ α′ < α, f(α0,α)(α′) =Xα′ = {xα′

1 , xα′2 , ...} ∈ V[U ]; if a skand e(α0,α) is the empty skand, then the

associated map f(α0,α) is constant and its image is equal to ∅ ∈ V[U .Conversely, for an arbitrary map f(α0,α) : (α0, α) → V[U ] we can as-

sociate a unique X(α0,α) such that associated map coincides with f(α0,α) :(α0, α) → V[U ].

Indeed, let f(α0,α) : (α0, α) → V[U ] be a map. We shall construct aunique skand X(α0,α) such that the associated map as above will coincidewith assumed map f(α0,α) : (α0, α) → V[U ]. The construction will bedone by transfinite induction. Consider the value f(α0,α)(α0) ∈ V[U ] off(α0,α) of its first argument α0 ∈ (α0, α) and the restriction f(α0+1,α) =f(α0,α)|(α0+1,α) of f(α0,α) on the subset (α0 +1, α) of (α0, α). We define nowa set X(0) whose elements are all elements of the set f(α0,α)(α0) ∈ V[U ]and the map f(α0+1,α). More precisely, X(0) is a discrete sum of the setf(α0,α)(α0) and the singleton Y(0) � f(α0+1,α), i.e. X(0) = f(α0,α)(α0) � Y(0).It is clear that a new set X(0) is nothing else than an initial map (function)f(α0,α) : (α0, α) → V[U ] because it uniquely reconstructs the map f(α0,α)

by putting f(α0,α)(α0) = X(0)\Y(0) and f(α0,α)(α′) = f(α0+1,α)(α′), α0 +1 ≤α′ < α.

(Note that a map or function is a special “functional relation” which canbe realized, written, denoted by in different ways like “a graph of function”,“implicit function”, etc., in our case of well-ordered domain (α0, α) it isdenoted by the set with elements of the image of its first argument and anelement which is the restriction of f(α0,α) : (α0, α) → V[U ] on (α0, α)), i.e.the map f(α0+1,α) : (α0 + 1, α) → V[U ]. In all realizations, representationsor designations of f(α′,α) by a graph of function, an implicit function or as inour case by a special recursive set X(α′), α0 ≤ α′ < α0 +ω, we speak aboutthe only mathematical object: a map or function f(α′,α) : (α′, α) → V[U ].)

In the same manner we define a unique set X(1) = Xα+1 � Y(1), whereY(1) � {f(α+2,α)} which uniquely reconstructs the map f(α0+1,α) and actu-ally equal to it. We continue the construction for all α′, i.e., change themap f(α′,α) by the set f(α′,α(α′) � Y(α′), Y(α′) � f(α′+1,α, α0 ≤ α′ < ω ≤ α.We obtain an extraordinary set X = X(0) � X(1) � X(2) � ... in the sense


of Mirimanoff:

X = {xα01 , xα0

2 , ..., {xα0+11 , xα0+1

2 , ..., {...{xα′1 , xα′

2 , ...}...}}}, (6)

α0 ≤ α′ < ω ≤ α.If α ≤ α0 + ω, then we stop our construction, and this Mirimanoff

extraordinary set is nothing else than the initial skand X(α0,α). But if α0 +ω < α, then in formula (6) we see only visible components and parametersof the initial function f(α0,α) : (α0, α) → V[U ]: f(α0,α)(α′), α′ ∈ (α0, α0+ω)and (α0, α0 + ω), respectively. And we do not see in (6) other hiddencomponents and parameters of the initial function: f(α0+ω,α)(α′), α0 +ω ≤α′ < α, and (α0 + ω, α), respectively.

We remember that each set X(n), 0 ≤ n < ω, is actually a map f(α0+n,α),and the latter is a pasting map f(α0+n,α) = f(α0+n,α0+ω) ∪ f(α0+ω,α). Thus,we have to add in (6) our description of f(α0+ω,α) by Mirimanoff sets X(ω) �X(ω+1) � ... � X(ωn) � ..., 0 ≤ n < ω, and so on ... up to the exhaustion ofall components f(α0,α)(α′), α0 ≤ α′ < α, and all parameters (α0, α). Thus,we obtain the need skand X(α0,α) in (5) with Xα′ = f(α0,α)(α′), α0 ≤ α′ < α.�

Remark 3. One can consider a rigid version of skand theory requiringϕ in Definition 2 to be an identical isomorphism. This skand theory cannaturally describe the “world of non-well-founded sets” given in [19], Chap.II, §5, which was “supposed to be of a highly artificial nature”. The formalsymbols x1, x2, ... were considered; and formally it was put that xn+1 ∈ xn

and ¬xi ∈ xj, i �= j + 1. Putting R(0) = {x1, x2, ...}, it was defined byrecursion R(α) as a power-set of the set

⋃β<α

R(β). For elements of the

set R(α) there was the following ∈-relation: if u is a set, then v ∈ u, ifv is an element of u; if u = xi, then v ∈ u, if v = xi+1. Notice that oneneeds in the above construction taken in [19] to identify xi with {xi+1},i ≥ 1. This abstract and formal construction can be naturally describedvia rigid skands as follows: put x1 = X(1,α) with Xα′ = ∅, 1 ≤ α′ < α,α ≥ ω; then xi = X(i,α) are restrictions of X(1,α) on (i, α), 1 < i < ω.And then form worlds of non-well-founded sets R(α), α ≥ 1, as above. Wewill not consider rigid skands in this paper because they do not allow us toinvestigate self-similar and reflexive sets.

Example 1.X(α0,α) = {{1, {2, {3, {...}}}}}, Y(α0,α) = {1, {{2, {3, {...}}}}}.Skands X(α0,α) and Y(α0,α) are not equal because the α0-component Xα0

of X(α0,α) is empty and the α0-component Yα0 of Y(α0,α) consists of oneelement, which is equal to 1. These skands have the same components, butin different order.

16 JU. T. LISICA

Remark 4. We shall postulate below that an arbitrary skand X(α0,α) ={x0

0, x01, ..., x

0λ0

, ..., {x10, x

11, ..., x

1λ1

, ..., {...}}} is considered as a set in V[U ]−

whose elements are sets or individuals x00, x

01, ..., x

0λ, ... of U and the skand

X(α0+1,α) = {x10, x

11, ..., x

1λ1

, ..., {...}};i.e., X(α0,α) = {x0

0, x01, ..., x

0λ0

, ..., X(α0+1,α)}. In particular, for l ≥ ω atrivial skand e(α0,α) can be considered as a set whose only element is this setitself, i.e., e(α0,α) = {e(α0,α)} because, by Definition 2, e(α0,α) = e(α0+1,α).So, Russell was not right saying that “X = {X} must be absolutely mean-ingless, not simply false”.

Clearly, if α = α0 + n, where n = 1, 2, ... are natural numbers, thenX(α0,α) = {x0

0, x01, ..., x

0λ0

, ..., {x10, x

11, ..., x

1λ1

, ...

..., {..., {xn−10 , xn−1

1 , ..., xn−1λn−1

, ...}}}}is an ordinary well-founded set whose elements are sets or individualsx0

0, x01, ..., x

0λ0

, ..., X(α0+1,α) of U.Example 2. X(0,ω) = {a0, {a1, {a2, {...}}}}, where ai, i = 0, 1, 2, ..., are

individuals or sets in U, where ω = ω0 is the first infinite ordinal.This is a skand of length ω or a two-element set in V−; i.e., X =

{a0, X(1,ω)}, where X(1,ω) = {a1, {a2, {...}}}. It is a typical example of anon-well-founded set, or an extraordinary set in the sense of Mirimanoff [43],[44], [45], where ai, i = 0, 1, 2, ..., are individuals; e.g., non-negative integers0, 1, 2, ...; i.e. X(0,ω) = {0, {1, {2, {...}}}}, or examples of Mirimanoff’scircular extraordinary sets of period n, where ai = ai+n, i = 0, 1, 2, ..., andn = 1, 2, ... is a fixed natural number; e.g., X(0,ω) = {0, {1, {0, {1, {...}}}}},where n = 2, a0 = 0 and a1 = 1.

The following is a more general example of a circular set.Example 3. Consider

X(0,ω) = {a0, a1, ..., aλ, ..., {b0, b1, ..bμ, ..., {a0, a1, ..aλ, ...

..., {b0, b1, ..., bμ, ..., {...}}}}}, (7)

where aλ and bμ are individuals or sets in U such that {a0, a1, ..., aλ, ...} and{b0, b1, ..bμ, ...} are elements of V[U ], 0 ≤ λ ≤ κ and 0 ≤ μ ≤ ν, respectively.In other words, an even component X2n is the set {a0, a1, ..., aλ, ...} and anodd component X2n+1 is the set {b0, b1, ..., bμ, ...}, 0 ≤ n < ω.

This skand can be considered as circular sets, i.e., X(0,ω) � X(1,ω) �X(0,ω) and X(1,ω) � X(0,ω) � X(1,ω) because X(0,ω) = {a0, a1, ..aλ, ..., X(1,ω)},X(1,ω) = {b0, b1, ..bμ, ..., X(2,ω)}, X(2,ω) = {a0, a1, ..aλ, ..., X(3,ω)} and, byDefinition 2, X(0,ω) = X(2,ω) and X(1,ω) = X(3,ω).

Example 4. Consider two skands

X(0,ω2) = {a0, a1, ..., aλ, ..., {b0, b1, ..., bμ, ..., {a0, a1, ..aλ, ...{a0, a1, ..., aλ, ..., {b0, b1, ..bμ, {...}}}...}}}, (8)


where aλ and bμ are individuals or sets in U, 0 ≤ λ ≤ κ and 0 ≤ μ ≤ν, respectively, where even components X2τ are equal to the fixed set{a0, a1, ..., aλ, ...} and odd components X2τ+1 are equal to the other fixedset {b0, b1, ..., bμ, ...}, 0 ≤ τ < ω2, and

Y(0,ω2) = {a0, a1, ..., aλ, ..., {b0, b1, ..., bμ, ..., {a0, a1, ..aλ, ...{b0, b1, ..., bμ, {a0, a1, ..aλ, {...}}}...}}} (9)

which can be obtained from X(0,ω2) by changing 2τ -components X2τ on theset {b0, b1, ..., bμ, ...} and 2τ+1-components X2τ+1 on the set {a0, a1, ..., aλ, ...},for all ω ≤ τ < ω2.

They can also be considered as circular sets; i.e., X(0,ω2) � X(1,ω2) �X(0,ω2) and X(1,ω2) � X(0,ω) � X(1,ω), becauseX(0,ω2) = {a0, a1, ..aλ, ..., X(1,ω2)}, X(1,ω2) = {b0, b1, ..bμ, ..., X(2,ω2)},X(2,ω2) = {a0, a1, ..aλ, ..., X(3,ω2)} and, by Definition 2, X(0,ω2) = X(2,ω2)

and X(1,ω2) = X(3,ω2) as well as Y(0,ω2) � Y(1,ω2) � Y(0,ω2) and Y(1,ω2) �Y(0,ω2) � Y(1,ω2), becauseY(0,ω2) = {a0, a1, ..aλ, ..., Y(1,ω2)}, Y(1,ω2) = {b0, b1, ..bμ, ..., Y(2,ω2)}, Y(2,ω2) ={a0, a1, ..aλ, ..., Y(3,ω2)} and, by Definition 2, Y(0,ω2) = Y(2,ω2) and Y(1,ω2) =Y(3,ω2). Nevertheless, X(0,ω2) �= Y(0,ω2), as well as X(1,ω2) �= Y(1,ω2). It isalso clear that X(0,ω) �= X(0,ω2) and X(0,ω) �= Y(0,ω2). �

Note that every well-founded set X = {x0, x1, ..., xλ, ...} can be consid-ered not only as a skand X(0,1) of length 1 but also as many other differentskands of different finite lengths in general. Indeed, fix, for example, oneelement of X . Let it be x0. Since X is well-founded we choose a de-scending ∈-chain x0 � x1

0 � x20 � ... � xn

0 such that xn0 is an individual

or the empty set, and fix it. Then X = X(0,n+1) = {x1, x2, ..., xλ, ..., x0},where x0 = {a1

1, a12, ..., a

1λ1

, ..., a10} and a1

0 = x10 = {a2

1, a22, ..., a

2λ2

, ..., a20},

a20 = x2

0 = {a31, a

32, ..., a

3λ3

, ..., a30},..., an−1

0 = xn−10 = {an

1 , an2 , ..., an

λn, an

0},an

0 = xn0 ; i.e.,

X = X(0,n+1) = {x1, x2, ..., xλ, ...

..., {a11, a

12, ..., a

1λ1

, ..., {..., {an1, a

n2 , ..., an

λn, ..., an

0}}}}.If for example, xn

0 = ∅, then there is another skand which gives the sameset X = X(0,n+2) = {x1, x2, ..., xλ, ..., {a1

1, a12, ..., a

1λ1

, ..., {......, {an

1, an2 , ..., an

λn, ..., {}}}}}.

Example 5. X(0,ω+1) = {a0, {a1, {a2, {...{aω}...}}}, where ai, i =0, 1, 2, ..., ω, are individuals or sets in U.

This is the simplest example of non-well-founded set whose only twoelements are a0 and skand X(1,ω+1) = {a1, {a2, {...{aω}...}}}. It was notconsidered by Mirimanoff because by remaining silent on the issue of anextraordinary set, i.e., X � X1 � X2 � ..., Mirimanoff seems to suggest

18 JU. T. LISICA

that it is in our terminology a skand of length ω only. Moreover, all suchskands and others of length ω +β, where β ∈ On, β ≥ 1, are here essentialgeneralizations and extensions of some of Mirimanoff’s extraordinary sets.�

3. The formal theory NBG[U ](1) of non-well-founded sets and itsmodel in the universe of elements U[U ](1)

Recall that we work here inside NBG− and NBG[U ]− set theories,where the former is the theory of “pure” sets and latter is the theory withindividuals U , and in both theories the Axiom of Foundation FA has beenomitted. As above, by V− and V[U ]− we denote the classes of sets inthese theories, as well as the universes of elements U− = V− and U[U ]− =V− ∪ U , respectively. Similar notations are for their extensions as thesame set theories plus the Axiom of Foundation FA: NBG and NBG[U ]set theories, where the former is the theory of “pure” sets and latter isthe theory with individuals U , V and V[U ] are the classes of sets in thesetheories, as well as the universes of elements U = V and U[U ] = V ∪ U ,respectively.

We want to extend NBG− and NBG[U ]− to NBG(1) and NBG[U ](1),respectively, by less restrictive axioms than FA, which will be based nev-ertheless on NBG

def= NBG(0). For transfinite inductive constructions we

also put NBG[U ]def= NBG[U ](0), V

def= V(0), V[U ]

def= V[U ](0), U

def= U(0),

U[U ]def= U[U ](0), and U def

= U (0), respectively.Suppose now that U (0) is a set (which can be empty) or class and we

will define a theory NBG[U ](1) which extend in these different cases NBG−and NBG[U ]−, turning our attention to the latter because the former isa particular case of NBG[U ]− when U = ∅. For this purpose we add toNBG(0) a wider class of individuals than U (0) by adding to it a new classof individuals U (1). Elements u

(1)λ ∈ U (1), λ ∈ Ord, are arbitrary skands

X(α0,α) of length l = (α − α0) ≥ ω taken with the forgetful operator Ewhich “forgets” the inner structure (∈ relations) of X(α0,α). More pre-

cisely, u(1)λ = EX(α0,α) and no element or object is a member of EX(α0,α).

Moreover, if X(α0,α) = X(β0,β), then EX(α0,α) = EX(β0,β) and they de-

fine the only individual u(1)λ ∈ U (1). We will need the inverse operator,

i.e., “remember operator” E−1, i.e., for any individual u(1)λ = EX(α0,α),

λ ∈ Ord, E−1u(1)λ = X(α0,α).

By NBG[U ](1) we understand the theory NBG[U ]− with the additionaxiom:


Skand Existence Axiom and Recognition Skand as a Set. Forany map f(α0,α) : (α0, α) → V[U ], α − α0 ≥ ω, there exists a unique skandX(α0,α) such that for each α′ ∈ (α0, α) one has Xα′ = f(α0,α)(α′). Thisskand is considered as a set X ∈ V[U ]− whose elements are elements ofthe set f(α0,α)(α0) and one more element X(α0+1,α) which is a restriction ofX(α0,α) on (α0 + 1, α), i.e.,

X = {xα00 , xα0

1 , ..., xα0λ , ..., X(α0+1,α)}. (10)

We will note this very set by the same symbol X(α0,α) but we shall differmeaning X(α0,α) as a skand (5) and X(α0,α) as a set (10).

Remark 5. One can notice that we ignore here slands X(α0,α) of fi-nite length, i.e., l = α − α0 < ω, because in this case the set X(α0,α) iswell-founded set and will not enrich V[U ](0). When a skand X(α0,α) has alength l = α − α0 ≥ ω, then the set X(α0,α) is a non-well-founded set be-cause by Skand Existence Axiom and Recognition Skand as a Set (shortlySEA&RSS) in X(α0,α), repeatting ω times, in the set X(α0,α) there is aninfinite ∈-descending chain:

X(α0,α) � X(α0+1,α) � X(α0+2,α) � ... � X(α′,α) � ..., α0 ≤ α′ < α. (11)

Since in NBG[U ](1) we postulate existence of new objects and sets wehave to revise the previous Extensionality Axiom of NBG[U ](0).

Strong Extensionality Axiom. Two sets (resp., classes) X (1) andY (1), whose elements are well-founded sets, individuals and skands as non-well-founded sets are equal if for each element x ∈ X (1) there is an elementy ∈ Y (1) such that x = y and for each element y ∈ Y (1) there is an elementx ∈ X (1) such that y = x, where “=” means the following:

1) the equality of individuals, when x, y ∈ U (0), i.e., (x = y) ⇐⇒(∀z)(x ∈ z ⇐⇒ y ∈ z);

2) the equality of well-founded sets, when x, y ∈ V[U ](0), i.e., by EA ofNBG[U ];

3) the equality of skands, when x, y ∈ E−1U (1), i.e., by Definition 2;4) the (iterative) equality of sets in V[U ](1), i.e., by using 1), 2), 3) in

SEA for a complex sets x, y ∈ V[U ](1).The latter class V[U ](1) is defined by transfinite recursion. Denote by

U ′ the discrete union U (0) � U (1). By formulas (2) and (3) we define aclass H[U ′] which is a class of well-founded sets and determines a modelof NBG[U ′], H[U ′] = U[U ′] and is an inner model of NBG[U ′]−. Now weapply the remember operator E−1 to all individuals of U (1) which are inany constituents of sets in V[U ′] = H[U ′] \ U (0) or individuals themselves

20 JU. T. LISICA

in H[U ′] \U (0), in short words, in all objects in X ∈ H[U ′] \ U (0) we changeall individuals of U (1) on the corresponding skands considered as sets, i.e.,each individual EX(α0.α) turns into a set X(α0.α).

One can see that after such changing V[U ′] turns into a class V[U ](1)

of sets which can be well-founded and not-well-founded. Moreover, byPropositions 1 and 2, V[U ](1) is a model of NBG[U ](1) and an inner modelof NBG[U ]−.

These definitions and construction of NBG[U ](1) give the following the-orem.

Theorem 1. The set theory NBG[U ](1) is consistent on the assumptionthat NBG− is consistent.

Proof. As we mentioned above, NBG− is consistent iff NBG[U ]− isconsistent. It is well known that if NBG[U ]− is consistent, then NBG[U ] =NBG[U ](0) is also consistent (see Exercise 4.86 in [41]). But NBG[U ] isconsistent iff NBG is consistent. Again NBG is consistent iff NBG[U ′] isconsistent. But the remember operator E−1 makes from NBG[U ′] a con-sistent theory NBG[U ](1), too. (See details in [47] or Chap. 4, Proposition4.50 in [41]).

Remark 6. In our case we have an additional class of individuals U (1)

as “pseudo-individuals” in the above sense, i.e., each ∈-descending chainis finite or terminates, i.e., reaches its bottom or “foundation” which is anempty set, or an individual, or a skand. �

Since ∅ ⊂ U it is clear that by the second part of SEA&RSS, i.e., RSS,we obtain the following embeddings:

V[U ] = V[U ](0) ⊂ V[U ](1) ⊂ V[U ]−. (12)

Examples 3, 4, 5, show that the former embedding is proper; we shall seethat the latter embedding is proper, too. �

4. Self-similar skandsThe following skands are of great interest in the study of the relation

X ∈ X in the Set Theory which we are going to clarify in this paper.Definition 3. A skand X(α0,α) is called self-similar if for each α1,

α0 ≤ α1 < α, there is an equality X(α0,α) = X(α1,α).It happens iff α = ωκ, where ω = ω0 is the initial countable ordinal,

κ ≥ 1, and each α′-component Xα′ of X(α0,ωκ), α0 ≤ α′ < ωκ, is the sameset {x1, x2, ..., xλ, ..., {.

Indeed, it is well known that each remainder ρ of an ordinal α �= 0 isequal to α if and only if α = ωκ, κ ≥ 0, ([36], Ch. VII, §7, Theorem 7).Recall that an ordinal ρ is called a remainder of an ordinal α if ρ �= 0 andthere exists an ordinal σ such that α = σ + ρ.


Then our assertion when skands are self-similar follows immediately fromDefinition 2.

Note that in the case κ = 0 the skands X(0,1) which are ordinary well-founded sets can be formally considered as “self-similar” skands of length1.

Recall also that ordinal numbers α of the form ωκ, κ ≥ 0, turn outto be the prime components or principal numbers of addition, i.e., ordinalnumbers α such that there is no decomposition α = β +γ where β < α andγ < α ([56], §19, Chap. XIV, Theorem 1, p. 323).

Example 6. X(0,ω) = {a0, {a0, {a0, {...}}}}, where a0 is an individualor set in U.

This is a self-similar skand of length ω or the simplest example of anextraordinary circular set of period 1.

More generally, X(0,ω) = {a0, a1, a2, ..., aλ, ...{a0, a1, a2, ..., aλ, ...{...}}},where aλ, 0 ≤ λ < κ, are individuals or sets in U, e.g., for λ = i, aλ = i,0 ≤ i < ω = κ, X(0,ω) = {0, 1, 2, 3, ..., {0, 1, 2, 3, ..., {...}}}.

Example 7. X(0,ω2) = {a0, {a0, {a0, {...}}}}, where a0, is an individualor a set in U.

This is an example of a self-similar skand of length ω2. We picture it inthe following figure:ω ↑

•{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, ...ωn ωn + 1 ωn + 2 · · · · · · ωn + m· · · · · · · · · ·

•{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, ...· α′ α′ + 1 · · · · · · ·· · · · · · · · · ·

•{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, ...ω2 ω2 + 1 ω2 + 2 · · · · · · ω2 + m•{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, ...ω ω + 1 ω + 2 · · · · · · ω + m

•{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, •{a0, ... −→0 1 2 . . . α0 α0 + 1 . m ω

Fig. 1.

and conclude that X(α0,ω2) ≈ X(α′,ω2), for all 0 ≤ α0 < α′ < ω2; i.e.,we see that X(0,ω2) is really a self-similar skand of length ω2. Noticealso that, by Definition 2, X(0,ω) = {a0, {a0, {a0, {...}}}} �= X(0,ω2) ={a0, {a0, {a0, {...}}}}.

Definition 4. A set X is called reflexive if X ∈ X .

22 JU. T. LISICA

It is not clear at once that reflexive sets do exist, in spite of Mirimanoff’s“ensembles de deuxieme sort” [43] and Eklund’s “Mengen, die Elementeihrer selbst sind” [22].

Consider now an “indeterminate” object X in V[U ]− of the followingform:

X = {x0, x1, x2, ..., xλ, ..., X}. (13)It is clear that the equation (13) is a general form of reflexive sets if they

exist.Proposition 3. Reflexive sets do exist. Moreover, there are a huge

number of different solutions (which form a proper class) of (13) in V[U ]−.Proof. Indeed, the following self-similar skands:

X(0,ωκ) = {x0, x1, x2, ..., xλ, ..., {x0, x1, x2, ..., xλ, ..., {...}}}, (14)

for each κ ∈ On, κ > 0, are solutions of (13) because, by the Axiom ofSkand Existence, objects in the form (14) do exist and by Definition 2,X(0,ωκ) = X(1,ωκ), and we obtain

X = X(0,ωκ) = {x0, x1, x2, ..., xλ, ..., X(1,ωκ)} = {x0, x1, x2, ..., xλ, ..., X}.(15)

�Remark 7. Proposition 3 shows that the relation X ∈ X is extremely

mulivalued and Russell was more or less right to call it “meaningless” be-cause without additional description it is undefined. One can say the samething about the relations X ∈ Y ∈ X which are also many-valued and,without additional description, are undefined, as Examples 3 and 4 tell us.

Remark 8. Many years ago the author noticed (better to say perceived)[38] that there are self-similar skands of length greater than ω and for along time has been thinking that only such skands of length ωκ, κ ≥ 2,are solutions of (13) which differ from each other and from the solution oflength ω, i.e., Mirimanoff’s extraordinary set solution. Now it is clear thatnot only, e.g.,

X(0,ω) = {x0, x1, x2, ..., xλ, ..., {x0, x1, x2, ..., xλ, ..., {...}}} (16)

andX(0,ω2) = {x0, x1, x2, ..., xλ, ..., {x0, x1, x2, ..., xλ, ..., {...}}} (17)

whose components are the same set Xα′ = {x1, x2, ..., xλ, ...}, 0 ≤ α′ < ωand 0 ≤ α′ < ω2, respectively, are different solutions of (13), but also, e.g.,

X(0,ω+1) = {x0, x1, x2, ..., xλ, ..., {x0, x1, x2, ..., xλ, {...{1}...}}} (18)

and

Y(0,ω+1) = {x0, x1, x2, ..., xλ, ..., {x0, x1, x2, ..., xλ, {...{2}...}}} (19)


are also different solutions of (13) because X(0,ω+1) = X(1,ω+1) and Y(0,ω+1) =Y(1,ω+1) and Xω = {1} �= {2} = Yω, although these solutions are not self-similar skands. So, reflexive sets need not be self-similar skands. On theother hand, all these different solutions above are isomorphic extraordi-nary sets in the sense of Mirimanoff [15], p. 40-41, and form a properclass. Moreover, they are identically isomorphic extraordinary sets in Mi-rimanoff’s sense and thus must be equal. So, identically isomorphic ex-traordinary sets, or equal extraordinary sets need not be equal in NBG(1).In other words, reflexive extraordinary sets in [43] as well as in [22] arenot well-defined, if we do not restrict them to skands of length ω. Judgingby his silence on the issue, it seems that Mirimanoff tacitly supposed thatthe length of the skands was equal to ω. Consequently, in Mirimanoff’sapproach we see only ω-phenomena and ignore trans-ω-phenomena. Thisis indeed the origin of the following error in logic which we are now goingto clarify. �

5. Applications to Russell’s paradox and its variants5.1 Russell’s paradox.In 1903 Russell published the famous paradox he had discovered two

years previously and of which he had informed other mathematicians bycorrespondence. Here is the original quotation: “We examined the contra-diction resulting from the apparent fact that if w be the class of all classeswhich as single terms are not members of themselves as many, then w as onecan be proved both to be and not to be a member of itself as many”([54],Chap. X, p. 107). Thus, in modern terminology, he defined the followingset:

R = {X | X /∈ X}, (20)where X are sets; i.e., R is the set of all sets that are not members ofthemselves, or the universal set formed by the property X /∈ X , which heand then all mathematicians considered to be a paradoxical set, or Russell’santinomy. Therefore, the set R was supposed to be inconsistent in Cantor’sNaıve Set Theory and the latter was called inconsistent, e.g., [9], p. 488-489.

The property or predicate X /∈ X was called Russell’s condition.Russell’s argument is the following:

R ∈ R ⇐⇒ R /∈ R (21)

which is a contradiction, and that is why the set R is inconsistent. Hence,R does not exist from Poincare’s point of view: “En mathematiques le motexister ne peut avoir qu’un sens: il signifie exempt de contradiction” ([52],

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p. 162), as well as from Russell’s: “The contradiction proves that the classas one, if it ever exists, is certainly sometimes absent” ([54], Chap. X, p.107).

Later on, when sets and proper classes were being distinguished and anypredicate (in particular, Russell’s condition) formed a class which existed byone of the class-formation axioms, it was supposed that Russell’s paradoxsaid nothing other than that R was a proper class, not a set. E.g., anexplicit exposition of it we take from [41], p. 239: “Let us verify now thatthe usual argument for Russell’s paradox does not hold in NBG−. By theclass existence theorem, there is a class Y = {x| x /∈ x}. Then (∀x)(x ∈Y ⇐⇒ x /∈ x). In unabbreviated notion this becomes (∀X)(M(X) =⇒(X ∈ Y ⇐⇒ X /∈ X)). Assume M(Y ). Then Y ∈ Y ⇐⇒ Y /∈ Y , which,by the tautology (A ⇐⇒ ¬A) =⇒ (A ∧ ¬A), yields Y ∈ Y ∧ Y /∈ Y . Now,by the deduction theorem, we obtain � M(Y ) =⇒ (Y ∈ Y ∧ Y /∈ Y ), andthen, by tautology (B =⇒ (A∧¬A)) =⇒ ¬B, i.e., the derived rule of proofby contradiction, we obtain � ¬M(Y ). Thus, in NBG−, the argumentfor Russell’s paradox merely shows that Russell’s class Y is a proper class,not a set. NBG− will avoid the paradoxes of Cantor and Burali-Forti ina similar way”. (In this citation we have only changed NBG into NBG−,which is in our notation the corresponding set theory; � A means that Ais a theorem.)

Now, by Proposition 3, we can state the following conjecture:Conjecture 1. In any set theory, if one assumes that R = {X |M(X)∧

X /∈ X} is a set, then the implication

R /∈ R =⇒ R ∈ R (22)

is false, i.e., Russell’s argument (21) is false and hence Russell’s paradox isnot a paradox in this set theory; thus, the proof that R is a proper class,not a set, via Russell’s paradox is incorrect.

The sense of this conjecture is the following: after assuming M(R) wenotice that

1) Russell’s condition X /∈ X becomes impredicative;2) by (13) and (14), the relation R ∈ R is multi-valued;3) a set R with the relation R ∈ R (which in this case is reflexive and

hence a non-well-founded set) in general is not well-defined, i.e., indefinite.We shall prove this conjecture for some well-known formal systems (set

theories), some from the syntactic and the semantic sides, some only fromsemantic side.

Proposition 4. Conjecture 1 is true in NBG, i.e., if M(R), then theimplication R /∈ R =⇒ R ∈ R is false in NBG.


Proof. It is well known that the axiom of regularity implies (∀X)(M(X)=⇒ X /∈ X). Indeed, otherwise, the set Y = {X} would not satisfy theaxiom of regularity Reg (4). (See the more general Proposition 4.44(c), p.279, in [41].) Thus, for each set X the statement (respectively, formula) X /∈X is always true (respectively, true in some and hence in all interpretationsor models of NBG) and the statement X ∈ X is always false (respectively,false in some, and hence in all interpretations or models of NBG) in NBG.In particular, if M(R), then R /∈ R is true and R ∈ R is false. Hence, by thecalculation of the truth function for implication, the following statement(respectively, formula) R /∈ R =⇒ R ∈ R is false (respectively, false in someand hence in all interpretations or models). �

Corollary 1. Russell’s paradox in NBG is not a paradox which “showsPr(R)”, and hence the classical proof that R, which in NBG coincideswith V[U ], is a proper class is not correct when it proceeds via Russell’sparadox.

Proof. Since by Proposition 4, the implication R /∈ R =⇒ R ∈ R isfalse in NBG we obtain that there is no equivalence (21) and hence theabove argument � M(R) =⇒ (R ∈ R ∧ R /∈ R) and, that by the derivedrule of proof by contradiction, one obtains � ¬M(R), does not work anymore. Thus, in NBG � ¬M(R) proved via Russell’s paradox is incorrect.

Nevertheless, the latter � ¬M(R) can be easily proved by the Maximal-ity Principle.

Proposition 5. Russell’s collection R = V[U ] in NBG is a properclass, not a set.

Proof. Since, by the axiom of regularity, for each set X ∈ V[U ] thestatement X /∈ X is true, we obtain the relation X ∈ R. Thus, R = V[U ].Assume that M(R). Then, as above, the statement R /∈ R is true. By theAxiom of Pairing, i.e., for every two sets X, Y there exists a set {X, Y }that has X and Y as its only members, in particular, when X = Y , thereexists a singleton {X}, thus, we obtain the singleton {R} ∈ V[U ]. Now, bythe Axiom of Union, we obtain the set R∪ {R} with a proper subset R,i.e., R ⊂ R∪{R}, which is in contradiction with the maximality of R, sinceit is the set of all elements X such that X /∈ X ; but by assertion M(R),R /∈ R. This proves the statement ¬M(R), i.e., Pr(R). �

Remark 9. In proofs of Propositions 4 and 5 we avoid the traditional(e.g., [41], Chap. 4, §1, [36], [12], [27], [4], etc.) arguments: AssumingM(R), by the derived rule of proof by definition (actually, by a hiddenaxiom) and that R ∈ R is a well-formed formula, the statement R /∈ R =⇒R ∈ R is supposed to be true. Indeed, the formula R ∈ R is well-formed,but in general a set R with the relation R ∈ R is not well-defined, because areflexive set R, i.e., R ∈ R, has a non-definite element R and we cannot say

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what set we are dealing with; moreover, if R is well-defined in some formalsystems, the statement R ∈ R can be false, as we have seen in the aboveproofs. If the statement R /∈ R =⇒ R ∈ R were true by the definition of Rand a well-formed formula R ∈ R or a well-defined non-well-founded set R,there should be a double-paradox of Russell. Indeed, in NBG there shouldbe two opposite statements: R /∈ R =⇒ R ∈ R is true and R /∈ R =⇒R ∈ R is false. Since the statement R ∈ R =⇒ R /∈ R is always true in alltheories, we conclude that in NBG, after assuming M(R), there is Russell’sparadox and there is no Russell’s paradox. Of course this double-paradoxwould imply that ¬M(R), but we proved the statement ¬M(R) via theMaximality Principle in a shorter and easier way, avoiding paradoxes andformal logical tricks with an impredicative definition of R under assumptionM(R), together with the well-formed or well-defined formula R ∈ R. Thereis another reason for us to avoid the derived rule of proof by definition. IfM(R), then by the Axiom of Power, there exists the set P(R) of allsubsets of R, and each element X ∈ P(R) satisfies Russell’s conditionX /∈ X . By Cantor’s theorem, the power of P(R) is greater than the powerof R, and thus, by Cantor-Bernstein’s theorem, P(R) is not a subset ofR. So, the statement M(R) =⇒ ¬(P(R) ⊆ R) is true. On the other hand,using the derived rule of proof, by definition we conclude that the statementM(R) =⇒ P(R) ⊆ R is true in spite of the fact that R ∈ P(R) and R /∈ R.We can use the definition of R for every element X of P(R) except X = Rto prove that X ∈ R, because all X ∈ P(R) are well-founded, and henceX /∈ X ; but for the well-founded set R we cannot conclude that R ∈ Rbecause a set R became indefinite. Notice that in NBG Russell’s “paradox”coincides with Cantor’s “paradox” as well as with Mirimanoff’s “paradox,”and after well-ordering R, it coincides with Burali-Forti’s “paradox” (seeall of them below).

Proposition 6. For any formal system (axiomatic set theory) K aninner model of which is the class V[U ] in NBG, after the supposition M(R)for R = {X |M(X)∧X /∈ X} the following statement (formula) R /∈ R =⇒R ∈ R is false.

Proof. It is known that for any theorem A of K, RelV[U ]A is also atheorem of K (see [41], Chap. 4, p. 282-283) or if A is a true statementin K, then it is true in any model (see Chap. 1, §, Theorem 1 in [19]). ByProposition 4, R /∈ R ⇐⇒ R ∈ R is false in inner model V[U ] of K, whereV[U ] is the universe in NBG of all well-founded sets. Hence it is false inK.

Corollary 2. R /∈ R =⇒ R ∈ R is false in NBG−.Proof. The same universe V[U ] in NBG is an inner model for NBG−.Corollary 3. R /∈ R =⇒ R ∈ R is false in ZF [U ] + (AFA).


Proof. Von Neumann’s universe V[U ] is an inner model for ZF [U ] +(AFA).

Corollary 4. R /∈ R =⇒ R ∈ R is false in Cantor’s Naıve Set Theory.Proof. We turn Cantor’s Naıve Set Theory into the following naıve

formal system by adding to the first order predicate caculus some naıveaxioms, judging by his silence on the issue, used without proving:

(I) Axiom of the singleton. If X is a set, then there exists a set {X}which contains X and only X as an element.

(II) Axiom of the union of two disjoint sets. If X and Y are twosets which have no common elements, then there exists a set X ∪ Y whoseelements are the elements of X and of Y , taken together.

(III) Axiom of the power set. If X is a set, then there exists a powerset P(X) whose elements are subsets of X .

(IV) Axiom of infinity. There exists an infinite set.Clearly, von Neumann’s universe V is an inner model for such a formal

system (naıve set theory) and for each X ∈ V the formula X /∈ X is alwaystrue and the formula X ∈ X is always false.

Proposition 7. Russell’s collection R in all the above set theories is aproper class, not a set.

Proof. Since in all these theories, after assuming M(R), the statementR ∈ R is false, we conclude that R /∈ R and, by the Maximality Principle,we obtain that R is not maximal. Thus, � M(R).

We can even avoid the definition of R. Indeed, assuming M(R), wenotice that von Neumann’s universe V is a subclass of R. Hence R \ (R \V) = V is a set which contradicts Proposition 5.

Thus, the set R is not inconsistent because of Russell’s famous paradox,or antinomy, as has been believed for more than one hundred years [61]but it is inconsistent because of a different argument, i.e., the MaximalityPrinciple.

Conjecture 2. In general, for any reflexive set X , there is no non-trivial predicate (well-formed) B(x) such that X = {x| B(x)}, in particular,B(X) = M(X)∧ X /∈ X does not define the reflexive set R under assump-tion M(R).

We say “in general” because in NBG−+(AFA) a reflexive set X ∈ X isuniquely determined by elements X \ {X}. We say “non-trivial predicate”because for each reflexive set X there is a rivial predicate (tautology):X = {x| x ∈ X}. The problem is that for each reflexive set X the nature ofits elements in X \ {X} is different from the nature of its reflexive elementX ∈ X . We know only one example in NBG−+(AFA) of a unique reflexiveset Ω such that Ω = {Ω}. �

5.2 Russell’s set is a parametric set.

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Later in [59], [28], [3] and [4] a parametric version of a real Russellian set(not a proper class) was introduced. For any set a ∈ V[U ]− (well-foundedor non-well-founded, it does not matter) the following set

Ra = {b ∈ a| b /∈ b} (23)

always exists as a set because it is an intersection of the proper class Rand the set a. As has been observed: “There is nothing paradoxical aboutRa. The reasoning that seemed to give rise to paradox only tells us thatRa /∈ a”. (See [4], p. 60). That means that the assumption Ra ∈ a givesthe paradox Ra ∈ Ra ⇐⇒ Ra /∈ Ra. This is the usual Russellian argument.Consider, e.g., the proof, given by Halmos in [28], p. 6, changing only thenotation, which we take from [3]. “Can it be that Ra ∈ a? We proceed toprove that the answer is no. Indeed, if Ra ∈ a, Ra ∈ Ra also (unlikely, butnot obviously impossible), or else Ra /∈ Ra. If Ra ∈ Ra, then, by (23), theassumption Ra ∈ a yields Ra /∈ Ra − a contradiction. If Ra /∈ Ra, then, by(23) again, the assumption Ra ∈ a yields Ra ∈ Ra − a contradiction again.This completes the proof that Ra ∈ a is impossible, so that we must haveRa /∈ a”.

In [59] Zermelo showed, by this relative Russell’s paradox, that the do-main B of his sets is not a set.

We see that here the implication Ra /∈ Ra =⇒ Ra ∈ Ra is false, too,as it was in Russell’s argument. If one, still believing in Russell’s paradox,uses the derived rule of proof by definition and states that the implicationRa /∈ Ra =⇒ Ra ∈ Ra is true, then the result Ra /∈ a follows logically fromthe double-paradox Ra /∈ Ra =⇒ Ra ∈ Ra is true and Ra /∈ Ra =⇒ Ra ∈ Ra

is false, since Ra ∈ Ra =⇒ Ra /∈ Ra is always true.Here is a shorter correct proof of this statement based on the Maximality

Principle.Proposition 8. Ra /∈ a.Proof. It is clear, that Ra /∈ Ra because otherwise, (i.e., Ra ∈ Ra), it

would be a member of a and, therefore, by (23), Ra /∈ Ra. Thus, we haveproved that Ra /∈ Ra. Notice the implication that Ra ∈ Ra =⇒ Ra /∈ Ra isalways true in similar situations.

Suppose now that Ra ∈ a. Then Ra∪{Ra} is a subset of a; moreover, itis a subset of a whose elements do not contain themselves, in particular, Ra,as we have proved above. Therefore, Ra is a proper subset of Ra ∪ {Ra};i.e., Ra ⊂ Ra ∪ {Ra} ⊂ a which, by (23), is in contradiction with themaximality of Ra, since it is the set of all elements b of a such that b /∈ b.Consequently, Ra /∈ a. �

Remark 10. In spite of the gap in Halmos’s proof, the author of thispaper cannot refrain from relating a witty observation made by Paul R.


Halmos (the author once met him and appreciated his wit as well as hisbrilliant lectures and books): “The most interesting part of this conclusionis that there exists something (namely Ra) that does not belong to a.The set a in this argument was quite arbitrary. We have proved, in otherwords, that nothing contains everything, or, more spectacularly, there isno universe. ‘Universe’ here is used in the sense of ‘universe of discourse,’meaning, in any particular discussion, a set that contains all the objectsthat enter into that discussion. In older (pre-axiomatic) approaches to settheory, the existence of a universe was taken for granted, and the argumentin the preceding paragraph was known as the Russell paradox. The moralis that it is impossible, especially in mathematics, to get something fornothing. To specify a set, it is not enough to pronounce some magic words(which may form a sentence such as ‘x /∈ x’); it is necessary also to have athand a set to whose elements the magic words apply”. (Ibid., p. 6-7).

5.3 Zermelo’s paradox.Zermelo founded his own paradox independently of Russell and said in

1908 that he had mentioned it to Hilbert and other people already before1903. Later, in 1936, in his letter to Scholz, he wrote that the set-theoreticalparadoxes were often discussed in the Hilbert circle around 1900, and hehimself had at that time given a precise formulation of the paradox whichwas later named after Russell (see [63]). We shall see below that Zermelo’sopinion was mistaken, and that Russell’s and his paradoxes are similar butnot the same. Zermelo’s paradox is the following: a set M that comprises aselements all of its subsets is inconsistent. Indeed, consider the set M0 of allelements of M which are not elements of themselves (e.g., the empty set isin M0) This set is a subset of M and hence by assumption on M , M0 ∈ M .If M0 ∈ M0, then M0 is not a member of itself. Hence M0 /∈ M0 and sinceM0 ∈ M , M0 ∈ M0: contradiction. (See [64], § 2.4; [61], p. 507). We noticethe same mistake: the implication (M0 /∈ M0 ∧ M0 ∈ M) =⇒ M0 ∈ M0 isfalse.

We can repeat said above, if one still believing in Russell’s paradoxuses the derived rule of proof by definition and states that the implication(M0 /∈ M0∧M0 ∈ M) =⇒ M0 ∈ M0 is true, then the result M0 /∈ M followslogically from the double-paradox (M0 /∈ M0 ∧ M0 ∈ M) =⇒ M0 ∈ M0 istrue and (M0 /∈ M0 ∧ M0 ∈ M) =⇒ M0 ∈ M0 is false.

Here is the solution of Zermelo’s paradox based on the Maximaliti Prin-ciple.

Proposition 9. There is no set M such that it comprises as elementsall of its subsets.

Proof. Suppose the contrary, and such set M exists. Then M0 = {X ∈M | X /∈ X} is well-defined and M0 ∈ M . If M0 ∈ M0 in any possible way,

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then M0 ∈ M0 ∈ M0 and, clearly, by definition of it, M0 /∈ M0. (The firstpart of Zermelo’s proof is correct.) By axiom (I), the singleton {M0} alsoexists and it is a subset of M , i.e., {M0} ⊂ M (because M0 ⊂ M and, bydefinition of M , M0 ∈ M). By axiom (II), we obtain that M0∪{M0} is alsoa subset of M and consists of elements which are not elements of themselvesbecause in addition to all elements of M0 with such a condition, M0 itselfsatisfied the same condition, i.e., M0 /∈ M0, as was proved in the first part.Clearly, M0 is a proper subset of M0 ∪ {M0}; i.e., M0 ⊂ M0 ∪ {M0} whichis in contradiction with the maximality of M0. Consequently, M doest notexist.

Notice also that contrary to Russell’s “set” R, which is not a set butdoes exist as a proper class, Zermelo’s set M does not exist as a proper classat all; in other words, there is no mathematical object such as Zermelo’sset M , but there is an object R in NBG− and in NBG. Moreover, inthe latter it is called the Universe V[U ] or the class of all well-foundedsets WF. That is why Zermelo’s and Russell’s paradoxes are meaningfullydifferent. M cannot be a proper class because in its rigid definition M hasto be an element of M , which is impossible for proper classes. If of courseone distinguishes sets and proper classes, then M is an object of set theory(a proper class) such that it comprises as elements all of its subsets (notproper subclasses) but M in this case is not defined in the original sense ofZermelo. �

6. Other set-theoretical paradoxesThe same argument is valid for the “set” On (Burali-Forti’s “paradox”

[13]), for the “set” Card of all cardinal numbers (Cantor’s “paradox,” firstmentioned in the letter to Dedekind of August 31. 1899, and later in [17]),for the “set” U (Hilbert’s “paradox” [61], p. 505), and for the “set” WF(Mirimanoff’s “paradox” [43]).

We shall also prove that the family NWF of all non-well-founded sets isnot a set but a proper class and it does not concern Russell-like paradoxes atall. As above all proofs will be based on Maximality Principle. Admittedly,it was implicitly used by Cantor in proving that the second number-classset is not countable [16], §16, [Theorem] D. His proof takes up at least onepage–more precisely, 32 lines). Now this proof requires only five lines.

Proposition 10. For each initial ordinal ωα, the set of all ordinals λsuch that λ < ωα cannot be of a power smaller than the power of ωα.

Proof. Indeed, if the power of the set X = {λ| λ < ωα} were smallerthan the power of ωα, then the power of the next ordinal number β =X ∪ {X} would be smaller than the power of ωα, too. Since β is greaterthan each ordinal λ in X , it is not a member of X . Hence X ⊂ X ∪ {β};


i.e., X is a proper subset of X ∪ {β}, which is in contradiction with themaximality of X . �

6.1 Burali-Forti’s paradox, or the antinomy of the greatest or-dinal.

The earliest antinomy in set theory was published in 1897 in [13]. It has,however, a non-trivial history, starting out from the fact that there wasnothing paradoxical in it, since in [13] there was no contradiction. This isbecause Burali-Forti had misconstrued Cantor’s definition of a well-oderedset, and used his own notion of a different kind of ordered sets which hecalled “perfectly ordered classes”, and proved that such classes are non-well-ordered. Russell [54], p. 323, reformulated the argument of Burali-Fortias a contradiction and gave it its present name. (This observation wasborrowed from [24], p. 306-307, and [61], p. 350). And we will finish thisstory with an amusing remark given by Halmos: “The contradiction, basedon the assumption that there is a set of all ordinals, is called the Burali-Forti paradox. (Burali-Forti was one man, not two.)” (See [28], p. 80.)We are going to show that Russell and others did not supply Burali-Forti’sparadox with a contradiction, although there are axiomatic systems wherethis contradiction arises.

Look at the modern explanation of Russell’s correction of Burali-Forti’santinomy of the greatest ordinal, e.g., in [61], p. 350. “Let Ω be the ordinalnumber of the well-ordered set of all ordinals, No. But, for every ordinalα, α + 1 > α. Thus Ω + 1 > Ω. But, for every α ∈ No, α ≤ Ω. ThusΩ + 1 ≤ Ω”.

Recall that, due to Cantor, “ordinal numbers” are ordinal types of “well-ordered aggregates”, i.e., well-ordered sets. Proper classes of equivalentwell-ordered sets can be represented by von Numann’s method [49], bychoosing a canonical set-representative of each proper class, that is to definean ordinal as a set α of sets X which is well-ordered by the relation ∈between its elements and transivity, i.e., if Z ∈ Y ∈ X , then Z ∈ X ,starting, e.g., with the empty set, i.e., ∅; {∅, {∅}}; {∅, {∅}, {∅, {∅}}}; ... . (Itwould perhaps have been fairer to say that the idea of this method for thefirst time was given by Mirimanoff [43], p. 46.) The collection of all ordinalsOn is well ordered by the relation ∈ (see [18], p. 8) and class transitive;i.e., if Z ∈ Y ∈ On, then Z ∈ On. Note that the class of all ordinals Noabove is in bijection with On; moreover, they are order isomorphic. Notealso that in NBG all ordinals are well-founded sets. Thus, for each X ∈ αwe have X �∈ X , in particular, α /∈ α, and moreover, we can even omit“well” in “well-ordered” (see, e.g., [41], Chap. 4, §5, Exercise 4.84). The

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situation in NBG− and in Naıve Set Theory is more complicated: “well-ordered” is sufficient and α ∈ α can happen. Nevertheless, we rememberthat in general the relations α ∈ α as well as α ∈ β ∈ α are indeterminate.

Proposition 11. On is a proper class, not a set.Proof. Let On be a collection of all transitive, well-founded sets ordered

by the relation ∈, i.e., we are in NBG. Suppose that On is a set. Thus,it is a transitive set well-ordered by the relation ∈ and hence is an ordinal.Then, by (I), {On} is also a set, and, by (II), On∪{On} is a set which isevidently well-ordered by the relation ∈ and transitive, too. Consequently,On ∪ {On} is an ordinal. (Notice that in NBG On /∈ On and {On ∪{On}} /∈ On because all sets in NBG are well-founded.) Moreover, by thesame argument, {On ∪ {On}} and On ∪ {On ∪ {On}} are sets and thusOn is a proper subset of On∪{On∪ {On}} because of the relation notedabove; i.e., On ∪ {On} /∈ On. The relation On ⊂ On ∪ {On ∪ {On}}is in contradiction with the maximality of On. Thus On is not a set. Itis a proper class since its existence is guaranteed by the predicate: to bean ordinal. If On is a family of all ordinal numbers represented in someother way, then there is an isomorphism ϕ : On → On. Thus, the relationsOn ∈ On and On + 1 ∈ On are impossible because by isomorphism ϕapplied to these relation the corresponding relations are false in On. �

Remark 11. Burali-Forti’s antinomy of the greatest ordinal is seen nowas two inconsistent inequalities Ω + 1 > Ω and Ω + 1 ≤ Ω. Consequently,we have the relations Ω < Ω+1 ≤ Ω in No and hence, by the isomorphismbetween No and On mentioned above, we obtain the relations Ω ∈ Ω ∪{Ω} ∈ Ω in On, which are indeterminate, as we have already seen manytimes. So in the classical argument, to conclude by definition that Ω+1 ∈ Ωis false. (The true state of affairs is that after the supposition that On = Ωis a set, the definition of On has been radically changed; i.e., each possiblerelation On ∈ On or Ω ∈ Ω∪{Ω} ∈ Ω changes the set On itself, and that iswhy On is indeterminate.) Consequently, when Russell supposedly showeda contradiction in the Burali-Forti example, he was wrong. If we supposethat On is a set, then naturally On ∪ {On} = Ω + 1 is an ordinal andΩ + 1 > Ω = On. But Ω ∪ {Ω} /∈ Ω, i.e., Ω + 1 �≤ Ω. Because Ω = On is awell-founded set in NBG, and in NBG− as well as in Naıve Set Theory, therelations Ω∪{Ω} ∈ Ω are indeterminate. Moreover, by the isomorphism ofclasses of all ordinals represented in any way, the above relations are falsein On, i.e., in the collection of all well-founded, transitive sets ordered bythe relation ∈. In our proof of the inconsistency of the set On we omitthis erroneous step On ∪ {On} ∈ On but add a new ordinal On ∪ {On}(different from each element of On) to On and obtain a larger set than On,and this is a real contradiction, because On is a collection of all ordinals.


6.2 Cantor’s paradox.The historically second set-theory antinomy belongs to Cantor: the set

V∗ of all sets (which was called later “Cantor’s paradise”) is inconsistentor paradoxical because the power set PV∗ is always of a greater cardinalitythan V∗, but at the same time PV∗ ⊆ V∗ since V∗ is the most inclusiveset of sets. Notice that the same mistake obtains: the relation PV∗ ⊆ V∗

implies, in particular, V∗ ∈ V∗ since V∗ ∈ PV∗. But the relation V∗ ∈V∗ is many-valued and indeterminate, as we saw above, and therefore,false. Thus there is no real paradox or contradiction in Cantor’s argument;moreover, in this case Cantor’s paradox is not a paradox at all. But whatwas correctly asserted via Cantor’s “paradox” was the following:

Proposition 12. V∗ is a proper class, not a set, or in Cantor’s de-nomination V∗ is an “inconsistent system”.

Proof. Suppose that V∗ is a set. Then, by the Power-set Axiom, thecollection (system) PV∗ of all subsets of V∗ is a set. It is clear, by (I), thatthere is an injection i : V∗ → PV∗, given by i(X) = {X}. On the otherhand, PV∗ is not a subset of V∗. Otherwise, there would be an embeddingj : PV∗ → V∗ and, by the Cantor-Bernstein theorem, PV∗ and V∗ wouldbe equivalent, which is false because of Cantor’s theorem |PV∗| > |V∗|.Thus, there exists an element X ∈ PV∗ such that X is a set and X /∈ V∗.Then, by (I), {X} is a set and, by (II), V∗ ∪ {X} is a set. Moreover,V∗ ⊂ V∗ ∪ {X}, i.e., V∗ is a proper subset of V∗ ∪ {X}, which is incontradiction with the maximality of V∗. Thus V∗ is not a set but a properclass, given by the predicate X = X , i.e., V∗ def

= {X | X = X & ∃{X}}and called the universal class (see [27], p. 124). (Of course, we could takeX = V∗ at the beginning, and since V∗ ∈ V∗ is indeterminate, concludethat V∗ /∈ V∗ and, by supposition that V∗ is a set, obtain, by (I) and (II),that V∗ is a proper subset of V∗ ∪ {V∗}. In other words, we could repeatour method of maximality; but we wished to find a mistake in Cantor’sargument.) �

Cantor himself reached the faulty conclusion that PV∗ ⊆ V∗, which isin contradiction with their cardinality, and obtained a supposed “paradox”.But it was only a presumption, which was proved to be false by Cantor’stheorem |PV∗| > |V∗|, nothing more. Therefore, there is no Cantor’sparadox. It would be a paradox if Cantor could prove that PV∗ ⊆ V∗.Meanwhile his conclusion PV∗ ⊆ V∗ was made via the Definition of V∗;but after the assumption that V∗ was a set, the Definition of V∗ wasradically changed, and V∗ ∈ V∗ became automatically many-valued. Inother words, Cantor had to prove that each set X ∈ PV∗ is an elementof V∗. He could do it by Definition of V∗ except in the one case when

34 JU. T. LISICA

X = V∗ ∈ PV∗, because in V∗ there are many (actually all) reflexive setsX including all skands X = X(0,α), α ≥ ω, such that X(0,α) = X(1,α). Then,since V∗ ∈ V∗ as Cantor concluded by Definition, a reflexive set V∗ shouldbe at once the skands of all lengths α ≥ ω, which is impossible. Thus, it isimpossible to prove that PV∗ ⊆ V∗.

Absolutely by a similar argument Cantor proved that the collectionCard of all cardinal numbers is not a set (August 31, 1899, letter toDedekind) and Cantor made the following conclusion: “... the system Cardis not a set. That is why there exist certain pluralities which are not at thesame time wholes (unities), i.e., pluralities for which the real ‘mutual beingof all their elements’ is impossible. They are those I call the ‘inconsistentsystems’; as for the rest I call them the ‘sets’ ” [62].

Here is a correct proof of Cantor’s statement.Proposition 13. Card is a proper class, not a set, or in Cantor’s

denomination Card is an “inconsistent system”.Proof. Suppose that Card is a set. It is known that Card is in a

bijection with the family In of all finite numbers together with the all initialordinals ωκ of On which we identify with the initial segments (0, ωκ) of On,κ ∈ On (if κ = 0, then the corresponding segment is the empty set). Sincewe supposed that Card is a set, therefore In is a set, by the ReplacementAxiom for sets, and we obtain that the discrete sum X =

∐ωκ∈In

[0, ωκ) is

also a set, by the Union Axiom for sets, and, by the Power-set Axiom,PX is a set. Hence there exists an initial ordinal ωλ such that PX and[0, ωλ) are bijective. It is clear that ωλ > ωκ, for each ωκ ∈ In (because ofCantor’s theorem: |P[0, ωκ)| > |[0, ωκ)|, κ ∈ On), which is in contradictionwith the maximality of In. Thus Card is not a set but a proper class, sinceCard ⊂ V∗. �

Remark 12. Notice that from the fact that On is not a set but a properclass, as has been proved above, it does not follow that its subclass Card isalso a proper class. On the other hand, there is a bijection between On andCard because there is an injection j : On → Card, given by j(κ) = ωκ,κ ∈ On, and we apply the Cantor-Bernstein theorem. Thus Card is not aset, but a proper class.

6.3 Hilbert’s paradox.At least earlier than 1905 Hilbert formulated a paradox of his own which

he had considered “purely mathematical” in the sense that it did not makeuse of notions from Cantor’s theory of cardinals and ordinals. Hilbert neverpublished the paradox: “I have never published this contradiction, but itis known to set theorists, especially to G. Cantor” [33], p. 204.


Let us quote it from [61], p. 505-506. “The paradox is based on aspecial notion of set which Hilbert introduces by means of two set formationprinciples starting from the natural numbers. The first principle is theaddition principle (Additionsprinzip). In analogy to the finite case, Hilbertargued that the principle can be used for uniting two sets together ‘intoa new conceptual unit’, a new set that contains the elements of both sets.This operation can be extended: ‘In the same way, we are able to uniteseveral sets and even an infinitely many into a union.’ The second principleis called the mapping principle (Belegungsprinzip). Given a set M, heintroduces the set MM of self-mappings (Selbstbelegungen) of M to itself.A self-mapping is just a total function (‘transformation’) which maps theelements of M to elements of M.

Now, he considers all sets which result from the natural numbers ‘byapplying the operations of addition and self-mapping an arbitrary numbertimes.’ By the addition priciple, which allows us to form the union ofarbitrary sets, one can ‘unite them all into a sum set U , which is well-defined.’ In the next step the mapping principle is applied to U , and weget F = UU as the set of all self-mappings of U . Since F was built fromthe natural numbers, by using the two principles alone, Hilbert concludesthat it has to be contained in U . From this fact he derives a contradiction.

Since there are ‘not more’ elements in F than in U there is an assignmentof the elements ui of U to elements fi of F such that all elements of fi areused. Now one can define a self-mapping g of U which differs from all fi.Thus, g is not contained in F . Since F contains all self-mappings, we havea contradiction. In order to define g, Hilbert used Cantor’s diagonalizationmethod. If fi is a mapping ui to fi(ui) = uf i

ihe chooses an element ugi

different from uf ii

as the image of ui under g. Thus, we have g(ui) = ugi �=uf i

iand g ‘is distinct from any mapping fk of F in at least one asssignment.’

Hilbert finishes his argument with the following obsorvation:“We could also formulate this contradiction so that, according to the

last consideration, the set UU is always bigger [of greater cardinality] thanF = U , but according to the former, is an element of F = U .” �

Let us comment on the above proof. In spite of the difference betweenCantor’s and Hilbert’s definitions of set (e.g., the empty set set is not inU , and other non-intersections) their proofs and arguments are similar andcontain the same mistake. Let us see it in Hilbert’s paradox. Hilbert hasproved, using Cantor’s diagonalization method, only that his suppositionthat F ⊆ U is false, nothing more. Thus, there is no contradiction in suchan unfortunate supposition. If he had proved that F ⊆ U was true, therewould be a real paradox to be named after him. What he has essentially

36 JU. T. LISICA

proved is that there is an element g of F such that it was not an elementof U . Then by his first principle U ∪ F is a set; moreover, U is a propersubset of U ∪ F because b is not in U and it is in U ∪ F . Thus, by theuniversality of U , (i.e., U is the set of all Hilbert’s sets), U is not a Hilbertset, nor is F = UU a Hilbert set. There is no paradox here at all. There isonly a proof by contradiction, and the supposition that U is a set is simplyfalse. �

6.4 Mirimanoff’s paradox.In [43], p. 43, Mirimanoff was actually concerned with the collection

WF of all well-founded sets, and presented his paradox:

WF ∈ WF ⇐⇒ WF /∈ WF (24)

as an analogue of Russell’s paradox. And he concluded that the set WFdid not exist, i.e., in Cantor’s terminology, it is an inconsistent system, or inmodern terminology, it is not a set. His argument is the following. It is clearthat WF /∈ WF because if it were not so and WF ∈ WF, then there wouldexist an infinite descending chain in WF, e.g., WF � WF � WF � ... andwe would obtain that WF is non-well-founded and hence WF /∈ WF.

On the other hand, if WF is non-well-founded; there exists an infinitechain x0 � x1 � x2... in WF for some x0 ∈ WF, and then x0 is evidentlynon-well-founded, which is wrong. Therefore, WF is well-founded. HenceWF ∈ WF; contradiction.

The last conclusion is false because WF ∈ WF is indeterminate as wesaw above more than once.

Here is a correct proof of the followingProposition 14. WF is a proper class, not a set.Proof. Assume that WF is a set. Then by (I) the singleton {WF}

is also a set. Since all elements of WF are well-founded we conclude thatWF is not an element of WF because if it were, then there would bean evident infinite descending ∈-sequence WF � WF � WF � ... ThusWF and {WF} are two sets without common elements. Then, by (II),X ′ = WF ∪ {WF} is also a set. Moreover, its elements are well-foundedand WF is a proper subset of X ′ because WF is a member of X ′ but nota member of WF. This is in contradiction with the maximality of WFbecause it is the well-founded universe, since WF is the collection of allwell-founded sets. Consequently, the assertion that WF is a set is false.By the well-founded predicate and the Existence Axiom for a class, WF isa proper class. �

In NBG(1) we have the following collection:

T = {X | X ∈ X} (25)


which is symmetric to Russell’s “set” R = {X | X /∈ X} [21], p. 277, and onthe assumption that it is a set it does not look paradoxical or inconsistent.We are not saying that T ∈ T, because otherwise, the relation T ∈ Tcannot even be called indeterminate as above; it is meaningless as Russelltells us in [58], p. 80. Indeed, T contains, e.g., all skands of arbitrarylengths; T ∈ T, if it were the case, would be a skand of a fixed length, andother skands of different lengths would be different from this fixed one. Inother words, T cannot be a skand of all lengths at once. Nevertheless,

Proposition 15. T is a proper class, not a set.Proof. Consider for an arbitrary well-founded set X , i.e., X ∈ WF,

a self-similar skand X(0,ω) with the components Xi = X , 0 ≤ i < ω, anddenote by L the subclass {X(0,ω)| X ∈ WF, Xi = X, 0 ≤ i < ω} whichconsists of all such self-similar skands X(0,ω) with Xi = X , 0 ≤ i < ω.Clearly, L ⊂ T. Since WF is a proper class then L, which is bijective toit, is also a proper class. Consequently, T is also a proper class because onthe contrary, if T were a set the relation L ⊂ T would imply that L werea set. �

All these results are well-known. What is new is the simple proofs ofthem and the unexpected discovery that the classical set-theoretical para-doxes are not paradoxes at all. Moreover, we maintain that the substanceof all such “paradoxical sets”, actually proper classes, is in the followingsimplest lemma of such a kind of proposition.

Lemma 1. The collection S of all singletons in WF is a proper class,not a set.

Proof. Suppose the contrary, and S is a set. By (I), {S} and {{S}}are also well-founded sets. Moreover, {S} /∈ S because all elements of Sare well-founded. Then, by (II), the disjoint union S ∪ {{S}} is also a setand S is a proper subset of S∪ {{S}} which contradicts the maximality ofS. �

Corollary 5. That propositions 4, 6-12 together with GeneralizedLemma 1, i.e., the collection of all singletons in V[U ]− form a proper class,not a set, is a consequence of Lemma 1.

The Proof is evident. One can easily find in all these cases that S is asubclass of the supposed “sets” or that there is a subclass of the supposed“sets” which is in bijection with S. Then the assumption that the widerclass is a set implies, by an axiom in NBG which says that the intersectionof any set X with an abitrary class Y is a set (see, [41], Chap. 4, Axiom S),that S is a set, contrary to Lemma 1. Indeed, the simplest proof is of theGeneralized Lemma 1, Proposition 12, Proposition 14, because evidentlythe class of all well-founded singletons is a subclass of all singletons aswell as S ⊂ V− and S ⊂ WF, and the statements made of them hold.

38 JU. T. LISICA

Since, by von Neumann’s axiom N, V− ≈ On there is a subclass of N, Onwhich is equivalent to S, and by the injection j : On → Card (Remark12), the statements of Propositions 11 and 12 hold. As to Russell’s set R(Proposition 5), it contains S as a subclass because, for each well-foundedset X ∈ R, the singleton {X} is also well-founded and {X} ∈ R; otherwise,{X} ∈ {X} and hence {X} = X and the latter is a non-well-founded set.In T there is a subclass of self-similar skands X(0,ω) whose componentsXi = X , 0 ≤ n < ω, X ∈ S, and thus Proposition 15 holds. �

Remark 13. In [4] there is another, also very short proof of a general-ized Lemma 1, but “from the top to the bottom”, whereas in our proof weare going “from the bottom to the top”. Here it is. If S in V[U ]− were aset, its union would be a set. But

⋃S contains the proper class V[U ]− of

all sets. This is because for all sets a, {a} ∈ S. Since⋃

S is both a properclass and a set, we have a contradiction. (See [4], p. 338, and note thatin [4] it has already been proved that the class of all sets is a proper class,although via Russell’s paradox; see Ibidem, p. 16). �

7. Comparison of NBG(1) with some other approaches to non-well-founded sets

First of all, notice that objects X (1) in V[U ](1) are essential extensionsof some but not all extraordinary sets in the sense of Mirimanoff [43], [44],[45] as well as non-well-founded sets in [22]. In [45], p. 33, Mirimanoffconsidered extraordinary sets in the following form:

E = {y, z, ...a, b, c, ...}, (26)

where sets y, z, ... depend themselves on E; in particular, he considered setsin the form

E = {E, a, b, c, ...}. (27)

In NBG(1) we consider not only the latter form and those which areself-similar skands, or circular skands of period 1, but also skands or extra-ordinary sets of the form

E = {a, b, c, ..., {E, x, y, z, ...}}, (28)

i.e., circular skands of period 2, and of course we consider also circularskands of arbitrary finite period n. But we do not consider extraordinarysets, or self-similar skands, e.g., of the form

E = {E, a, b, c, ..., {E}} (29)

which is a particular case of (26). This restriction of ours is done knowinglyand it is similar to the restriction of well-founded sets in comparison with


non-well-founded sets which can be called 0-rank self-similar skands (well-founded sets), 1-rank self-similar skands (circular sets of period 1), and wecould postulate the existence of self-similar skands of arbitrary finite rank.For our purposes we need such a restriction. So, skands are not extensionsof all of Mirimanoff’s extraordinary sets of the form (26).

On the other hand, the definition (26) provides the possibility of theexistence of skands whose lengths are greater than ω. However nothingwas said in [45] about such things, and it appears that Mirimanoff tacitlysupposed that the length of skands was equal to ω. Only a single footnote onpage 12 in [22] says that “∈μ-constituent of relation (i.e., ∈-chains of lengthμ) can be defined for an arbitrary ordinal number μ, but we consider in thispaper only finite numbers μ”. Nevertheless, this footnote, given by Elkund,does not determine the fact that non-well-founded objects, presented in [22],are skands of ω-length in our terminology, and hence non-well-defined. So,skands in NBG(1) are essential extensions of extraordinary sets of lengthω. �

Now we want to compare NBG(1) with Aczel’s model or theory of non-well-founded sets [1] which was motivated by Robin Milner’s work [42]on computer science modeling of concurent processes. Aczel’s model ofso-called hypersets was successfully applied to the treatment of the Liarparadox [3] and various other vicious circle phenomena [4].

Starting out with the Zermelo-Frankel set axiomatic system ZFC, whichincludes the axiom of choice and the axiom of foundation, Aczel rejected thelatter and proposed his own Anti-Foundation Axiom (AFA for short);and with a natural correction of the Axiom of Extensionality he ob-tained the set axiomatic system ZFA− + AFA, which is consistent on theassumption that ZFC is consistent.

The binary relation x ∈ y Aczel pictures as an element of a graph x −→ ywith nodes x, y and edge (x, y) between them; finite x0 −→ x1 −→ ... −→xn−1 −→ xn and infinite sequences finite x0 −→ x1 −→ x2 −→ ... aswell as pointed and accesible graphs, labelled graphs, children, decorationsof graphs, etc., are understood in the usual way. So, every set with an∈-relation can be pictured by labelled graphs.

Then the Labelled Anti-Foundation Axiom says: Every labelledgraph has a unique labelled decoration.

This axiom is a natural extension of Mostowski’s Collapsing Lemma thattells us that every well-founded labeled graph has a unique decoration inWF. On the other hand, AFA is a strong restriction of V[U ]− becauseof Scott’s axiom: For any extensional relation R on A and for any notR-well-founded x ∈ A there is no set containing all possible images of x

40 JU. T. LISICA

under isomorphisms between (A, R) and a transitive structure (T,∈). (Un-published paper of 1960). As a consequence of this axiom the uniquenessof a “Mostowski collapse” cannot be consistently postulated for non-well-founded sets. (See details in [26].)

There is an equivalent formulation of AFA which is nearer to our papercalled the Solution Lemma.

The Anti-Foundation Axiom: Every flat system of equations has aunique solution, [4], p. 72.

It is enough for us because a particular example of a flat equation is(13). Thus, axiomatic systems ZFA−+AFA and NBG(1) are incompatible,although they have mutual objects such as the empty skand e(0,ω) = {{...}}in our paper and Ω = {{...}} in [1]. Moreover, the hyperset Ω∗ = {Ω, Ω∗} isequal to Ω because Ω = {Ω} = {Ω, Ω} and we apply the Solution Lemma.We will see below that the generalized skand {e(0,ω), {e(0,ω), {...}}} is notequal to e(0,ω).

Analysing AFA we see that a unique solution of a flat equation (13) isactually a proper class of all its solutions in V[U ](1). Thus the quotientclass V[U ](1) recieved by the solution lemma equivalent relation, we obtaina subclass of the hyperset universe. That is why Russell’s paradox andother set-theoretical paradoxes (except Cantor’s and Hilbert’s paradoxes)are possible in the hyperset universe (and only in such systems) of course,on the assumption that R or other considered classes are sets, and, whatis especially important, that one has to refer to AFA in implications suchas, e.g., R /∈ R =⇒ R ∈ R. �

8. ν-generalized skands, formal theory NBG[U ](ν) and its modelin the universe of elements U[U ](ν), and the universe U[U ](Ω)

We have already extended the formal set theory NBG[U ]− by a newaxiom SEA&RSS which is less restrictive than FA and have obtainedthe formal theory NBG[U ](1) as well as have enriched V[U ] = V[U ](0) bynew objects and have extended it to the class V[U ](1) of sets X which canbe either well-founded or non-well-founded the latter are skands of lengthl ≥ ω considered as sets of V[U ]. Moreover, we have shown that theclass U[U ](0) = V[U ](0) ∪ U is a model of NBG[U ](1) and an inner modelof NBG[U ]−. We can successively continue this process of constructionfor each ordinal number ν ∈ On, defining formal theories NBG[U ](ν

′),1 < ν′ ≤ ν, enriching V[U ](0) by new ν′-generalized skands, 1 < ν′ ≤ ν.Moreover, there are sequential embeddings:

V[U ](0) ⊂ V[U ](1) ⊂ V[U ](2) ⊂ ... ⊂ V[U ](ν) ⊂ V[U ]−, (30)


where all embeddings are proper.This can be done by the following transfinite induction.Suppose that we have already constructed NBG[U ](ν

′) for each 1 ≤ν′ < ν, and classes V[U ](ν

′) of sets in the coresponding theories, enrichingthem by ν′′-generelized skands as sets of V[U ]−, 1 < ν′′ ≤ ν′, and hence theuniverse of elements U[U ](ν

′) = V[U ](ν′)∪U of this theory NBG[U ](ν

′). Nowwe construct a formal set theory NBG[U ](ν) and the class V[U ](ν) of all setsin NBG[U ](ν) and hence the universe of elements U[U ](ν) = V[U ](ν) ∪ Uof this theory NBG[U ](ν), and the following sequential embeddings areproper:

V[U ] = V[U ](0) ⊂ V[U ](1) ⊂ V[U ](2) ⊂ ... ⊂ V[U ](ν′) ⊂ V[U ]− , (31)

0 ≤ ν′ < ν.Definition 5. Let f(α0,α) : (α0, α) → ⋃

0≤ν′<ν

V[U ](ν′) be a map. By

a ν-generalized skand X(ν)(α0,α)

we understand a usual skand as a system ofembedded curly braces {α0{α0+1...{α′...α′}...α0+1}α0}, indexed by ordinalsα′ ∈ (α0, α), whose pairs of braces {{ and {} are not filled or filled by previ-ously defined sets of

⋃0≤ν′<ν

V[U ](ν′), i.e., whose components X

(ν)α′ coincide

with the images f(α0,α)(α′), respectively, α0 ≤ α′ < α.

Definition 6. Two ν-generalized skands X(ν)(α0,α) and Y

(ν)(β0,β) are equal if

the segments (α0, α) and (β0, β) are isomorphic as well-ordered sets and thecorresponding components X

(ν)α′ = Y

(ν)β′ , where this unique isomorphism is

given by ϕ : (α0, α) → (β0, β) and β′ = ϕ(α′), α0 ≤ α′ < α, β0 ≤ β′ < β,are equal as sets of

⋃0≤ν′<ν

V[U ](ν′).

ν-Generalized Skand Existence Axiom and Recognition ν-Gene-ralized Skand as a Set.

For any map f(α0,α) : (α0, α) → ⋃0≤ν′<ν

V[U ](ν′), α−α0 ≥ ω, there exists

a unique skand X(ν)(α0,α) such that for each α′ ∈ (α0, α) one has X

(ν)α′ =

f(α0,α)(α′). This ν-generalized skand X(ν)(α0,α) is considered as a set X ∈

V[U ]− whose elements are elements of the set f(α0,α)(α0) and one more

element X(ν)(α0+1,α) which is a restriction of X

(ν)(α0,α) on (α0 + 1, α), i.e.,

X = {xα00 , xα0

1 , ..., xα0λ , ..., X

(ν)(α0+1,α)

}, (32)

42 JU. T. LISICA

where {xα00 , xα0

1 , ..., xα0λ , ...} is a set whose elements are in the class

⋃0≤ν′<ν

V[U ](ν′). The only element of the set X is the skand X

(ν)(α0+1,α)

con-

sidered as a set is not an element of⋃

0≤ν′<ν

V[U ](ν′) but an element of

V[U ]−.

Notice that ν-genelized skands X(ν)(α0,α) of a length l = (α − α0) < ω are

elements of the class⋃

0≤ν′<ν

V[U ](ν′) and are out of our interest.

As above we will note this very set by the same symbol X(ν)(α0,α)

but we

shall differ X(ν)(α0,α)

as a ν-generalized skand in Definition 5 and X(ν)(α0,α)

asa set (32).

As in the case of the first step of induction we want to extend NBG−and NBG[U ]− to NBG(ν) and NBG[U ](ν), respectively, by less restrictiveaxioms than FA, which will be based nevertheless on NBG.

As above we suppose that U (0) is a set (which can be empty) or class,then we will define a theory NBG[U ](ν) which is an extension of theoriesNBG[U ]− and NBG−; the latter in the case U = ∅. For this purposewe add to NBG(0) a wider class of individuals than U (0) by adding toit a new class of individuals U (ν). Elements u

(ν)λ ∈ U (ν), λ ∈ On, are

arbitrary skands X(ν)(α0,α) of length l = (α−α0) ≥ ω taken with the forgetful

operator E which “forgets” the inner structure (∈ relations) of X(ν)(α0,α).

More precisely, u(1)λ = EX

(ν)(α0,α)

and no element or object is a member

of EX(α0,α). Moreover, if X(ν)(α0,α)

= X(ν)(β0,β)

, then EX(ν)(α0,α)

= EX(ν)(β0,β)

and they define the only individual u(ν)λ ∈ U (ν). As above we need the

inverse operator, i.e., “remember operator” E−1, i.e., for any individualu

(ν)λ = EX

(ν)(α0,α), λ ∈ On, E−1u

(ν)λ = X

(ν)(α0,α).

Since in NBG[U ](ν) we postulate existence of new objects and sets wehave to revise the previous Extensionality Axiom (sortlyEA) of NBG[U ](0).

ν-Strong Extensionality Axiom.

Two sets (resp., classes) X (ν) and Y (ν) whose elements are well-foundedsets, individuals and ν-generalized skands as non-well-founded sets areequal if for each element x ∈ X (ν) there is an element y ∈ Y (ν) suchthat x = y and for each element y ∈ Y (ν) there is an element x ∈ X (ν) suchthat y = x, where “=” means the following:




3) the equality of ν-generalized skands, when x, y ∈ E−1U (ν), i.e., byDefinition 6;

4) the (iterative) equality of sets in V[U ](ν), i.e., by using 1), 2), 3) inν-SEA for a complex sets x, y ∈ V[U ](ν).

The latter class V[U ](ν) is defined by transfinite recursion. Denote byU ′ the discrete union U (0) � U (ν). By formulas (2) and (3), we define aclass H[U ′] which is a class of well-founded sets and determines a modelof NBG[U ′], H[U ′] = U[U ′] and is an inner model of NBG[U ′]−. Now weapply the remember operator E−1 to all individuals of U (ν) which are inany constituents of sets in V[U ′] = H[U ′] \ U (0) or individuals themselvesin H[U ′] \ U (0), in short words, in all sets X ∈ H[U ′] \ U (0) as well asindividuals in U(ν) ⊂ X ∈ H[U ′] \ U (0) we change all these individuals onthe corresponding ν-generalized skands and consider them as sets like in(??).

One can see that after such changing, V[U ′] turns into the class V[U ](ν)

of all sets in this theory NBG[U ](ν) which can be well-founded and non-well-founded. Moreover, by Propositions 1 and 2, V[U ](ν) is a model ofNBG[U ](ν) and an inner model of NBG[U ]−.

This construction gives the following theorem.Theorem 2. The set theory NBG[U ](ν) is consistent on the assumption

that NBG− is consistent.Proof is the same as in Theorem 1.In our case we have an additional class of individuals U (ν) as “pseudo-

individuals” in the above sense, i.e., each ∈-descending chain is finite orterminates, i.e., reaches its bottom or “foundation” which is an empty set,or an individual, or a skand, or ν′-generalized skand, 1 < ν′ ≤ ν. �

Since by definitions and construction of U ′ we obtain the following em-beddings:

U = U (0) ⊂ U (1) ⊂ U (2) ⊂ ... ⊂ U (ν) = U ′. (33)

It is clear that by (33) and the second part of ν-SEA&RSS, i.e., νRSS,we obtain the following embeddings:

V[U ] = V[U ](0) ⊂ V[U ](1) ⊂ V[U ](2) ⊂ ... ⊂ V[U ](ν) ⊂ V[U ]−. (34)

Note that the latter embedding is proper because of the possibility ofthe next step ν + 1 of trasfinite induction. �

44 JU. T. LISICA

At last, since we have constructed NBG[U ](ν) and V[U ](ν) for all ν ∈ Onwe obtain the following proper embeddings:

V[U ] = V[U ](0) ⊂ V[U ](1) ⊂ V[U ](2) ⊂ ... ⊂ V[U ](ν) ⊂ ... ⊂ V[U ](Ω) ⊂ V[U ]−,(35)

where V[U ](Ω) =⋃

ν∈On

V[U ](ν) and the formal non-well-founded sets theory

NBG[U ](Ω) has all axioms of NBG[U ]− plus two additional axioms. Weshall start with the following definitions

Definition 7. Let f(α0,α) : (α0, α) → V[U ](Ω) be a map. By a Ω-

generalized skand X(Ω)(α0,α)

we understand a usual skand as a system ofembedded curly braces {α0{α0+1...{α′...α′}...α0+1}α0}, indexed by ordinalsα′ ∈ (α0, α), whose pairs of braces {{ and {} are not filled or filled by sets ofV[U ](Ω), i.e., whose components X

(Ω)α′ coincide with the images f(α0,α)(α′),

α0 ≤ α′ < α.Definition 8. Two Ω-generalized skands X

(Ω)(α0,α)

and Y(Ω)(β0,β)

are equal ifthe segments (α0, α) and (β0, β) are isomorphic as well-ordered sets and thecorresponding components X

(Ω)α′ = Y

(Ω)β′ , where this unique isomorphism is

given by ϕ : (α0, α) → (β0, β) and β′ = ϕ(α′), α0 ≤ α′ < α, β0 ≤ β′ < β,are equal as sets of V[U ](Ω).

Ω-Generalized Skand Existence Axiom and Recognition Ω-Gene-ralized Skand as a Set.

For any map f(α0,α) : (α0, α) → V[U ](Ω), α − α0 ≥ ω, there exists

a unique skand X(Ω)(α0,α) such that for each α′ ∈ (α0, α) one has X

(Ω)α′ =

f(α0,α)(α′). This Ω-generalized skand X(Ω)(α0,α)

is considered as a set X ∈V[U ]− whose elements are elements of the set f(α0,α)(α0) and one more

element X(Ω)(α0+1,α)

which is a restriction of X(Ω)(α0,α)

on (α0 + 1, α), i.e.,

X = {xα00 , xα0

1 , ..., xα0λ , ..., X

(Ω)(α0+1,α)

}, (36)

where {xα00 , xα0

1 , ..., xα0λ , ..., X

(Ω)(α0+1,α)

} is a set whose elements are mebmers

of the class V[U ](Ω).Notice that in this limit case each Ω-generalized skand X

(Ω)(α0,α)

, α−α0 ≥ω, coincides with ν-generalized skand X

(ν)(α0,α) for some ordinal ν ∈ On

and thus X(Ω)(α0+1,α) is not a new element which should be enreach V[U ](Ω).

Indeed, it is a consequence of the following proposition.


Proposition 16. For any Ω-generalized skand X(Ω)(α0,α)

there exists a

minimal ordinal ν and a unique ν-generalized skand X(ν)(α0,α) such that for

each α′ ∈ (α0, α) one has X(Ω)α′ = X

(ν)α′ .

Proof. Since, by Definition 7, each X(Ω)α′ is a set and X

(Ω)α′ = f(α0,α)(α′),

α0 ≤ α′ < α, and X(Ω)α′ ∈ V[U](Ω) =

⋃ν′∈On

X (ν′) we can find a min-

imal ordinal ν′α′ such that f(α0,α)(α′) ∈ V[U ](�

′α′). Indeed, V[U ](�

′α′) is

the intersection of all V[U ](�′′) such that f(α0,α)(α′) ∈ V[U ](�′′). Since{να0, να0+1, ..., να′, ...} is a set which contains α ordinals we can take ν′′ =

supα′∈(α0,α)

ν′α′ which always exists. Then putting ν = ν′′ + 1, we obtain the

conclusion of Proposition 16.Moreover, Ω-GSEA&R Ω-GSS is the weakest restriction axiom among

all previous axioms FA, SEA&RSS, 1-SEA&RSS, ...,ν-SEA&RSS,...1 < ν ∈ On. Hence V[U ](Ω) is the largest class of wel-founded and non-well-founded sets in the theory NBG[U](Ω). We also denote by U[U ](Ω) =V[U ](Ω) ∪ U the universe of elements or as usually say “the universe ofarguments”.

Ω-Strong Extensionality Axiom. Two sets (resp., classes) X (Ω) andY (Ω) whose elements are well-founded sets, individuals and Ω-generalizedskands as non-well-founded sets are equal if for each element x ∈ X (Ω) thereis an element y ∈ Y (Ω) such that x = y and for each element y ∈ Y (Ω) thereis an element x ∈ X (Ω) such that y = x, where “=” means the following:



3) the equality of Ω-generalized skands, when x, y ∈ E−1U (Ω), i.e., byDefinition 8;

4) the (iterative) equality of sets in V[U ](Ω), i.e., by using 1), 2), 3) inΩ-SEA for a complex sets x, y ∈ V[U ](Ω).

Theorem 3. The set theory NBG[U ](Ω) is consistent on the assumptionthat NBG− is consistent.

Proof is the same as in Theorem 1.In our case we have an additional class of individuals U (ν) as “pseudo-

individuals” in the above sense, i.e., each ∈-descending chain is finite orterminates, i.e., reaches its bottom or “foundation” which is an empty set,or an individual, or a skand, or ν-generalized skand, for some 1 < ν ∈ On.�

46 JU. T. LISICA

9. Skand operations and categories of generalized skandsLet α ≥ ω be an arbitrary fixed ordinal number. We consider now

all Ω-generalized skands X(Ω)(0,α) whose components X

(Ω)α′ , 0 ≤ α′ < α, are

elements of U(Ω) = V(Ω), i.e., we consider the formal non-well-founded settheory NBG(Ω) without individuals, i.e., U = ∅ or, in other words, thetheory of “pures” sets. (For simplicity, we omit the index (Ω) and writeX(0,α) instead of X

(Ω)(0,α)

, Xα′ instead of X(Ω)α′ , and generalized skand instead

of Ω-generalized skand.)It is clear, that all such skands form a class (proper class), not a set.

We denote it here by S(α). Inside S(α) we define the following so calledskand-operations.

Definition 9. By the union X(0,α) ∪ Y(0,α), intersection X(0,α) ∩ Y(0,α),difference X(0,α) \ Y(0,α) of two elements X(0,α) and Y(0,α) of S(α) we un-derstand a generalized skand Z(0,α) whose components Zα′ are Xα′ ∪ Yα′ ,Xα′ ∩ Yα′ and Xα′ \ Yα′ , respectively, 0 ≤ α′ < α. By the power of ageneralized skand X(0,α) we understand a generalized skand PX(0,α) whosecomponents (PX)α′ are PXα′.

Definition 10. By the mapping f(0,α) : X(0,α) → Y(0,α) of two general-ized skands X(0,α) and Y(0,α) we understand a generalized skand f(0,α) whosecomponents fα′ are the mappings fα′ : Xα′ → Yα′ , where Xα′ and Yα′ arecomponents of X(0,α) and Y(0,α), respectively, 0 ≤ α′ < α. Moreover, f(0,α)

is an injection, surjection and bijection, respectively, if the same are all map-pings fα′ , 0 ≤ α′ < α. In particular, a generalized skand X(0,α) is a subskandof Y(0,α), if there exists an identical injection 1(0,α) : X(0,α) → Y(0,α).

It is clear that one can define other skand-theoretical operations suchas products and coproducts, inverse and direct limits, pull-backs and push-outs, the equivalence relation and quotients, etc., by the set-operations onthe corresponding components of skands, respectively. We shall not usethese constructions here, and thus omit details.

One can easily verify that the elements of S(α) as objects and mappingsbetween such objects as morphisms form a category which we denote bySk(α).

Of special interest for us here are categories Sk(ωκ)(P ), κ ≥ 0, whoseobjects are self-similar generalized skands X(0,α)(X), i.e., when α = ωκ,κ ≥ 0 is fixed, and whose components are permanent, i.e., Xα′ = X , for each0 ≤ α′ < α, X ∈ V[U ](Ω) and whose morphisms are self-similar generalizedskand-mappings f(0,α)(f) of such generalized skands whose components arepermanent, i.e., mappings fα′ = f : X → Y , for each 0 ≤ α′ < α.

For κ = 0, we consider the usual category of sets whose objects areelements of V[U ](Ω); i.e., sets and morphisms are mappings of sets.


Proposition 17. For each X(0,ωκ)(X) ∈ Sk(ωκ)(P ), κ ≥ 1, there is anindex λ, 1 ≤ λ, such that X(0,ωκ)(X) ∈ U (λ).

Proof. Since X ∈ V[U ](Ω) there exists an ordinal number ν ≥ 0 suchthat X ∈ V[U ](ν). Among all such ν there is a minimal number and wedenote it also by ν. Then X(0,ωκ)(X) ∈ V[U ](ν+1) and we put λ = ν + 1.

It is clear that all skand-operations on self-similar generalized skandsgive us a self-similar generalized skand as a result. We shall often use thefollowing skand-operation:

Definition 9. Let X(0,ωκ)(X) (X(0,ωκ) for short) be a self-similar gen-eralized skand, where κ ≥ 1 is fixed. By a singleton-skand we understandthe following self-similar generalized skandS(0,ωκ)(X(0,ωκ)) = {X(0,ωκ), {X(0,ωκ), {...}}}.

It is clear that, by Definition 6, the empty skand e(0,ωκ) is not equal to thesingleton-skand S(0,ωκ)(e(0,ωκ)) in contrast to their equality in ZFA−+AFA(see [1], p. 8) as we promised to show in paragraph 7.

There is an evident functor Fκ′κ′′ : Sk(ωκ′)(P ) → Sk(ωκ′′

)(P ), 1 ≤ κ′ ≤κ′′, which associates with every object X(0,ωκ′)(X) the object X(0,ωκ′′)(X)and with every self-similar morphism f(0,ωκ′

)(f) : X(0,ωκ′)(X) → Y(0,ωκ′

)(Y )the self-similar morphism f(0,ωκ′′)(f) : X(0,ωκ′′)(X) → Y(0,ωκ′′)(Y ). A verifi-

cation of the category’s axioms is trivial. Moreover, all categories Sk(ωκ)(P ),κ ≥ 1, are isomorphic to each other and hence to the usual category of sets,which are elements of U(Ω) and morphisms are maps of such sets.

Proposition 18. The skand V(0,ωκ) which is a skand-union of all self-similar generalized skands of length ωκ, i.e., V(0,ωκ) =

⋃

X∈V[U ](Ω)

X(0,ωκ)(X),

where κ ≥ 1 and fixed, is not an element of V[U ](Ω); i.e., it is a proper class,not a set.

Proof. Suppose the contrary, and V(0,ωκ) is a set. Consequently, thereexists a generalized singleton-skand S(0,ωκ)(V(0,ωκ)) of V(0,ωκ) which is a set,too. Thus, there exists a singleton skand S(0,ωκ)(S(0,ωκ)(V(0,ωκ))). SinceS(0,ωκ)(V(0,ωκ)) is not an element of a permanent component of V(0,ωκ)

(otherwise, we should have a skand of the formV(0,ωκ) = {a, b, c, ..., {V(0,ωκ)}, V(0,ωκ)} which is impossible) we obtain thatV(0,ωκ) is a proper subskand of V(0,ωκ) ∪ S(0,ωκ)(S(0,ωκ)(V(0,ωκ))) which isin contradiction with the maximality of V(0,ωκ). Consequently, V(0,ωκ) isnot a set, but a proper class, and there is no a generalized singleton-skandS(0,ωκ)(V(0,ωκ)) of it, which is of a great importance for us. �

10. A new representation of ordinal and cardinal numbers

48 JU. T. LISICA

We recall once more that, due to Cantor, an ordinal number α (shortlyan ordinal or an element of On) is “the ordinal type of a well-ordered set”[34], p. 152. Likewise, a cardinal number ℵα (shortly a cardinal or anelement of Card) was defined by Cantor as “the power type of equivalentsets” [34], p. 87. In other words, an ordinal number α is a common symbolfor the class of all isomorphic (“similar”) well-ordered sets and a cardinalnumber (n ≥ 0, for natural or finite cardinal numbers and ℵν, ν ∈ On,ν ≥ 0, for transfinite or infinite cardinal numbers) is a common symbol forthe class of equivalent sets, i.e., which are into a one-to-one correspondence.There is a natural binary relation < between two ordinals (respectively,cardinals) α and β iff there exist well-ordered sets (respectively, sets) Aand B of the ordinal (respectively, cardinal) types α and β, respectively,and an initial segment B′ ⊂ B such that A and B′ are similar (respectively,equivalent). (Note that we identify here cardinal numbers with the naturaland initial ordinal numbers.)

There are diffferent represenations of ordinals numbers as special well-founded well-ordered sets, e.g., initial segments (0, α) of On ordered byinclusion (Cantor [15]); the canonical representation of ordinals by puresets ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, etc. (Mirimanoff [43]); the represe-nation of ordinal numbers by well-founded sets which are ordered by ∈ andtransitive (von Neumann [49]). There is a series of similar constructionswhich represent the class of all ordinal numbers (and hence all cardinalnumbers) by non-well-founded sets which are well-ordered by ∈ and tran-sitive. The state of affairs is that these constructions give one more detailconcerning the last ordinal and cardinal number, called here the eschaton.

Definition 10. A reflexive set (in particular, a self-similar skand) α,which elements are also reflective sets, is called an ordinal number if itis reflexively well-ordered by the relation ∈ between its elements, i.e., forX, Y ∈ α, X ∈ Y includes the case X = Y since X ∈ X ; moreover, it istransitive, i.e., if X, Y, Z ∈ α and Z ∈ Y ∈ X , then Z ∈ X ; it satisfies thefollowing conditions: 1) for every X, Y ∈ α, if X �= Y , then either X ∈ Yor Y ∈ X , but not both; 2) every non-empty subset of α has a minimalelement. Two ordinal numbers α and β are equal if there exists a bijectionbetween them, which is an order-preserving function.

Remark 14. Definition 10 differs from the analogous classical definitionof an ordinal number in NBG because in the latter case sets X, Y, Z arewell-founded, as opposite to the former; one more remark: if α ∈ β, thenα < β and there is no relation α ∈ α in the classical case, contrary toDefinition 10, where α ∈ β implies in general α ≤ β, since an equalitymay exist in the case β = α. One can compare the definition of a reflexive


well-ordering with a reflexive total ordering R in [41], p. 9, such that everynon-empty subset of the field of R has an R-least element.

The classes Onωκ , κ ≥ 1, of all ordinals in the sense of Definition 10 canbe defined by the following transfinite induction.

Let κ > 0 be a fixed usual ordinal number, i.e., an element of On. Webegin with the empty skand e(0,ωκ) = {{{...}}} and call it the first elementof Onωκ , denoting it by e(1). Using Definition 9, we put successively

e(1) = e(0,ωκ),e(2) = {e(1), {e(1), {...}}}= {e(1), e(2)},e(3) = {e(1), e(2), {e(1), e(2), {...}}} = {e(1), e(2), e(3)},.............................e(n) = {e(1), e(2), ..., e(n−1), {e(1), e(2), ..., e(n−1), {...}}}=

= {e(1), e(2), ..., e(n−1), e(n)},.............................e(ω) = {e(1), e(2), e(3), ..., {e(1), e(2), e(3), ..., {...}}}=

= {e(1), e(2), e(3), ..., e(ω)},e(ω+1) = {e(1), e(2), e(3), ..., e(ω), {e(1), e(2), e(3), ..., e(ω), {...}}} ={e(1), e(2), e(3), ..., e(ω), e(ω+1)},..................................... .It is clear thate(1) ∈ e(1) and e(1) ⊆ e(1), e(1) ∈ e(2) and e(1) ⊂ e(2), e(1) ∈ e(3) and

e(1) ⊂ e(3), ... e(1) ∈ e(ω) and e(1) ⊂ e(ω), ...;e(2) ∈ e(2) and e(2) ⊆ e(2), e(2) ∈ e(3) and e(2) ⊂ e(3), e(2) ∈ e(4) and

e(2) ⊂ e(4),... e(2) ∈ e(ω) and e(2) ⊂ e(ω),...;and so forth.One can see that each ordinal e(α) is the ordinal type of the 0-component

e(α)0 of the skand e(α). Moreover, although there is no the ordinal number

0 among self-similar skands, there is a one-to-one correspondence betweenthe class of all ordinals Onωκ = {e(1), e(2), ..., e(α), ...}, α ≥ 1, and the classof all α ∈ On, because the ordinal type of 0-component e(α), which is nota self-similar skand, is the same as the initial segment (0, α) of On. �

Consider now a more interesting description of the class of all ordinalsin the sense of Definition 10. It is the skand-union of all ordinal numbers,i.e., Ω(0,ωκ) = { ⋃

α≥1e(α), { ⋃

α≥1e(α), {...}}}.

Proposition 19. The self-similar skand Ω(0,ωκ), where κ ≥ 1 and isfixed, is not an element of V[U ](Ω); i.e., it is a proper class, not a set.

Proof. Suppose the contrary, and Ω(0,ωκ) is a set. Consequently, thereexists a generalized singleton-skandS(0,ωκ)(Ω(0,ωκ)) = {Ω(0,ωκ), {Ω(0,ωκ), {...}}} of Ω(0,ωκ) which is a set, too.

50 JU. T. LISICA

Thus, there exists a singleton skandS(0,ωκ)(S(0,ωκ)(Ω(0,ωκ))). Since S(0,ωκ)(Ω(0,ωκ)) is not an element of a per-manent component of Ω(0,ωκ) (otherwise, we should have a skand of theform ω(0,ωκ) = {a, b, c, ..., {Ω(0,ωκ)}, Ω(0,ωκ)} which is impossible) we obtainthat Ω(0,ωκ) is a proper subskand of Ω(0,ωκ)∪S(0,ωκ)(S(0,ωκ)(Ω(0,ωκ))) whichis in contradiction with the maximality of Ω(0,ωκ). Consequently, Ω(0,ωκ) isnot a set, but a proper class, and there is not a generalized singleton-skandS(0,ωκ)(Ω(0,ωκ)) of it, which is of great importance for us. �

Thus Ω(0,ωκ) with a fixed κ ≥ 1 is a generalized skand-class whose per-manent component is a well-ordered class Onωκ .

Moreover, formally Ω(0,ωκ) ∈ Ω(0,ωκ), which is not in contradiction withan agreement that classes are not elements of classes. In our case, with aspecific definition of skand-operations, Ω(0,ωκ) cannot be an element of anyclass or set, e.g., the singleton S(0,ωκ)(Ω(0,ωκ)) which does not exist at allas an element of V[U ](Ω), but, by our natural construction, it is an elementof itself.

We see also that Ω(0,ωκ′) and Ω(0,ωκ′′) are isomorphic for every 1 ≤ κ′ <

κ′′ and we can omit indexes; i.e., we write Ω and call this generalized skand-class the last ordinal number or the eschaton in the sense of Definition 10,because it is well-ordered by ∈, and is class-transitive. In other words, Ω isa common symbol for the ordinal type of well-ordered proper classes whoseall initial segments are sets. It is the last, indeed, because there are nomore units, i.e., generalized singleton-skands one could add to Ω.

It is clear that Ω is the initial class-ordinal number because it is notequinumerous to any smaller ordinal number. Indeed, any α < Ω is aset and hence is not equivalent to Ω. By definition, the cardinality |A|of any proper class A is defined as the unique class-cardinal Ω which isequinumerous to A (the existence of such equinumerousness follows fromthe well-ordering theorem).

Proposition 20. Ω is a strongly inaccessible class-cardinal, not a setcardinal.

Proof. Let e(ων) < Ω, where ων is the initial ordinal number. Thenthe power-skand Pe(ων) is a set, and hence its permanent component isnot in one-to-one correspondence with the permanent component Ω0 ofΩ(0,ωκ), κ ≥ 1 is fixed; thus, Pe(ων) < Ω. Moreover, for any e(ωα) < Ω andβ < Ω, the sum of cardinals, i.e., the initial ordinals,

∑α<β

e(ωα) is a set and

hence its permanent component is not in one-to-one correspondence withthe permanent component Ω0 of Ω(0,ωκ), κ ≥ 1 is fixed and hence Ω is notits ordinal type. �


Remark 15. The eschaton Ω looks like the initial ordinal ω which isalso a strongly inaccessible cardinal with respect to all finite numbers, andis the first transfinite ordinal. The same can be said for Ω, which is astrongly inaccessible class-cardinal with respect to all infinite numbers, i.e.,all transfinite numbers, and is the first trans-infinite class-ordinal, or thefirst trans-definite ordinal, as it was called in [6].

Definition 11. By a proper generalized skand-class we understandX(α0,α), at least one component Xα′, α0 ≤ α′ < α, α ∈ On of which isa proper class, in particular, a proper generalized self-similar skand-class.

Definition 12. By a hyper-skand and generalized hyper-skand we un-derstand X(α0,Ω), whose components Xα′, α0 ≤ α′ < Ω, are elements ofV[U ] and V[U ](Ω), respectively, in particular, a self-similar hyper-skandand generalized self-similar hyper-skand X(α0,Ω)(X).

Thus O(0,ω) = {0, 1, 2, ..., ω, ω + 1, ..., {0, 1, 2, ..., ω, ω + 1, ..., {...}}},C(0,Ω) = {0, {1, {..., {n, {...{ω, {ω + 1, {...{α, {...}}}}}}}}}} andE(0,Ω) = {e(1), {e(2), ..., e(n), {...{e(ω), {e(ω+1), {...{e(α), {...}}}}}}}}}},

α ∈ On, are examples of a skand-class, a hyper-skand and a generalizedhyper-skand of all ordinals, respectively.

Definition 13. By a proper (generalized) hyper-skand-class we under-stand X(α0,Ω), at least one component Xα′ , α0 ≤ α′ < Ω of which is aproper class, in particular, a proper (generalized) self-similar hyper-skand-class.

Remark 16. Definitions 11, 12, 13 are very general in the sense thatdefined objects are outside of the U−-world and are not even subclassesof it. Nevertheless, there are operations similar to operations on properclasses which are subclasses of V[U ](Ω). On the other hand, there are nosuch operations as the power-skand, singleton-skand or other set-theoreticoperations. It is indeed true: “The content of a concept diminishes asits extension increases; if its extension becomes all-embracing, its contentmust vanish altogether”. We need these definitions for descriptions of someaspects of the Skand Theory considered here. �

11. Applications to ε-numbersWhat we now want to show are applications of skands outside of the

Skand Theory considered above.Remark 17. The concept of a skand, i.e., objects X(α0,α) above, is

wider than its concrete realization as a system of embedded braces and thecomponents thereof; it is not rigidly attached to curly brackets and it mayalso be a system of embedded round brackets, e.g., streams

s = (a1, (a2, (a3, (....(aω, (aω+1, (...(aλ, (aλ+1, (...)))))))))), (37)

52 JU. T. LISICA

where aλ ∈ A, A is a set, 1 ≤ λ < Λ ∈ On, (see, in particular, the case of acountable system of embedded round brackets in [4], p. 34-35, 197-208), or asystem of embedded angle brackets for the set theoretic operation modelingthe operation of ordered pairs, etc. Skands X(0,α) can be interpreted as alimit power or continued exponential with a basis which is an ordinal γ0 ≥ 1such that X0 = γ0 is the first component of X(0,α) and with the exponentX(1,α), i.e.,

X(0,α) = γX(1,α)

0 (38)such that Xα′ = γα′ (notice that here we do not differ a singleton {γα′})which is exactly Xα′ with its element itself γα′, the more so, as it is anindividual), γα′ ∈ On, γα′ > 0, 0 ≤ α′ < α, i.e., a transfinite sequence ofbasis-exponents

X(0,α)sign= γ∧

0 γ∧1 γ∧

2 ...∧γ∧ωγ∧

ω+1...∧γα′ ...

def= γ

X(1,α)

0 = ... = γγ

γ...γ..

.

α′2

10 .

(39)

Note only that the latter term in (39) should be properly defined (seebelow), and in (39) it is only a designation, nothing else.We shall use such a skand-exponent in application to the theory of ε-numbers in the sense of Cantor. For convenience in the further designationof such a skand-exponent we prefer braces {{, i.e., E(0,α) = {γ0, {γ1, {...}}}to the power-sign ∧ in (39) or expressions like [γ0, γ1, γ2, ...γω, γω+1, ..., γα′, ...]in [51], as a descending sequence of two-element non-well-founded setsE(0,α) � E(1,α) � ... � E(α′,α) � ..., where E(α′,α) = {γα′, E(α′+1,α)},0 ≤ α′ < α.

Let γ, ξ be arbitrary ordinal numbers such that γ > 0 and ξ ≥ 0. Werecall that the ξth power of γ, i.e., γξ, is defined by the following transfiniteinduction:

γ0 = 1, (40)

γα+1 = γαγ, (41)

γλ = limα<λ

γα, (42)

for limit-numbers λ = limα<λ

α.

And as in arithmetic, γ is called the basis, ξ, the exponent, of the powerγξ.

Consider now the following equation:

γξ = ξ (43)


with indeterminate ξ.The roots ξ = α of the equation (43) in the case γ = ω and α < ω1,

where ω1 is the smallest non-denumerable ordinal, Cantor called epsilon-numbers. More precisely, “to distinguish them from all other numbers Icall them the “ε-numbers of the second number-class” ([16], §20).

We can omit these restrictions of Cantor’s, since all his results on theε-numbers are valid in general cases. Here we repeat Cantor’s constructionin a generalized form.

Let γ > 0 be a fixed ordinal. If α > 0 is any ordinal number whichdoes not satisfy the equation (43), it determines an increasing sequence αn,0 ≤ n < ω, by means of the equalities

α0 = α, α1 = γα α2 = γα1, ..., αn = γαn−1, ... . (44)

Then limn

αndef= sup

nαn = E(α) of this increasing sequence always exists

because On is well-ordered, and we call it an ε-number, too.Indeed, in the trivial case, when γ = 1, the only root of the equation

(43) is evidently ξ = 1, and hence this actually increasing sequence in (44)is the constant sequence αn = 1, 0 ≤ n < ω, and thus lim

nαn = sup

nαn =

E(α) = 1, for every α > 1.If γ > 1, then (44) is an ascending sequence (in Cantor’s terminology,

an “ascending fundamental series”) because

γ > 1 =⇒ γα ≥ α, (45)

for every α ≥ 0 (see, e.g., [36], Chap. VII, §6); in our case, when α > 1 anddoes not satisfy the equation (43), we have γα > α and, by (45), γγα

> γα,and so on. Consequently, (44) is an ascending sequence; indeed, for all0 < n < ω, α1 > α0, α2 > α1, α3 > α2, ..., αn > αn−1, ... .

Put now E(α) = limn

αn = supn

αn which is a limit-ordinal number and

always exists, because for the set A = {α0, α1, ..., αn, ...}, there exists anordinal β < β0 (for some fixed β0), which is greater than each element ofA (see [36], Chap. VII, §2, Theorem 6). Consequently, the least of such β,which always exists in the well-ordered set (0, β0) since each of its subsetshas the smallest element, is the desired ordinal number E(α).

By (41), (42), the function f(α) = γα is ascending and continuous;therefore, we have γE(α) = γ

limn

γαn

= limn

γαn = limn

αn+1 = E(α); i.e.,

E(α) satisfies (43).Cantor considered the case γ = ω, α = 1 and proved that

E(1) = limn

ωn, (46)

54 JU. T. LISICA

whereα0 = 1 α1 = ω α2 = ωα1, ..., αn = ωαn−1, ... , (47)

is an ε-number [16], §20, [Theorem] A. Moreover, ε0 = E(1) = limn

ωn is the

least of all the ε-numbers ([16], §20, [Theorem] B). (This, of course, is truein his own sense; in our general construction there are two more ε-numbers1, when γ = 1 and ω, when 1 < γ < ω and α = γ, which are evidentlysmaller than ε0.)

He also showed that after the least ε-number, ε0, there follows then thenext greater one:

ε1 = E(ε0 + 1), (48)and so on; i.e., there is the following formula of recursion:

εn = E(εn−1 + 1), (49)

1 ≤ n < ω. ([16], §20, [Theorem] D).The limit lim

nενn of any ascending sequence εν0 , εν1, ..., ενn, .. of ε-number

ενn , 0 ≤ n < ω, is an ε-number, too. ([16], §20, [Theorem] E). Finally, allthe totality of ε-numbers of the second number-class is a well-ordered set

ε0, ε1, ..., εn ... εω, εω+1, ... εα′ , ... (50)

of the second number-class type, and has thus the power ℵ1, where 0 ≤α′ < Ω, ([16], §20, [Theorem] F); i.e., Ω is the initial ordinal, in recentterminology, which is the first after the initial ordinal ω. (Here we quotenotations ωn, n ≥ 1, and Ω in [34], pp. 196, 199, literally; thus, do notconfuse them with ωκ, κ = 1, 2, ..., and Ω below, respectively, which denoteabsolutely different objects.)

We also mention two more of Cantor’s results. If ε′ is any ε-number,ε′′ is the next greater ε-number, and α is any number which lies betweenthem:

ε′ < α < ε′′, (51)then E(α) = ε′′ ([16], §20, [Theorem] C); and if ε is any ε-number and α isany number such that 1 < α < ε, then ε satisfies the three equations:

α + ε = ε; αε = ε; αε = ε (52)

([16], §20, [Theorem] G).Note that for the ε-number ω in our general construction, for each α

such that 1 < α < ω, (52) also holds.All these results of Cantor are valid in the general case of course, with

natural corrections: ordinal types, the cardinality of the initial ordinals,etc. But we want more. We want with the help of self-similar skands toclarify this general situation.


Cantor’s formula (46) of the least (in Cantor’s sense) ε-number ε0 canbe symbolically written in the following form:

ε0 = ωωω...ω

...

; (53)

i.e., by misuse of language, “ε0 is the power of ω whose exponent is thepower of ω, whose exponent is the power ω, etc., more precisely, ω times‘the power of’ ”, or “the first ‘limit power’ of ω whose exponents at eachnth place of the skand-exponent E(0,ω) is ω, 1 ≤ n < ω”. How else?

Let us denote (53) by a bit shorter formula:

ε0 = ωωω...ω

...

= E(0,ω)(ω)def= ωE(1,ω)(ω), (54)

where X(0,ω)(ω) denotes the self-similar skand-exponent of length ω whosecomponents En, 0 ≤ n < ω, are ω (see Remarks 1, 17). Actually, wewant to generalize this notation and the notion of the “limit power” to thefollowing one:

εκ = E(0,α)(γ)def= γE(1,α)(γ) = γγγ..

.γ..

.γ...γ

...

, (55)

for γ > 1, α = ωκ, κ ≥ 1, and κ ∈ On.Actually, we have to explain the meaning of the symbol E(0,α)(γ) =

γγγ...γ

...γ..

.γ..

.

as a good way of describing all possible ε-numbers because,by Definition 2, E(0,ωκ)(γ) = E(1,ωκ)(γ) and therefore,

γE(0,ωκ)(γ) = γE(1,ωκ)(γ) = E(0,ωκ)(γ); (56)

i.e., E(0,ωκ)(γ) in (55) is a root of the equation (43), for an arbitrary ordinalκ ≥ 1, and hence is an ε-number.

Here is an explicit explanation of this idea.Definition 14. By a skand-exponent E(α0,α) of length l = α − α0 ≥ ω

we understand a system of embedded curly braces, indexed by α′ ∈ (α0, α),all of whose components are one-element, moreover, for each α0 ≤ α′ < α,Eα′ = {γα′{ or Eα′ = {γα′}, if α′ = α − 1, where γα′ �= 0 and γα′ ∈ On. Ifall components are equal to γ, then we write E(α0,α)(γ) = {γ, {γ, {...}}} andconsiner it as a descending sequence of a two-elements non-well-founded setE(α0,α) � {γ, E(α0+1,α)} � {... � {γ, E(α0+α′,α)} � ..., 0 ≤ α′ < α − α0.

56 JU. T. LISICA

Definition 15. Two skand-exponents E1(α0,α) and E2

(β0,β) are calledequal if the segments (α0, α) and (β0, β) are isomorphic as well-orderedsets, where ϕ : (α0, α) → (β0, β) is this isomorphism, and the correspondingcomponents E1

α′ and E2β′ are equal, for each β′ = ϕ(α′), α0 ≤ α′ < α.

Definition 16. By an ω-limit power of the skand-exponent

E(α0,ω) = {γα0, {γα0+1, {...}}} = γγ..

.

α0+1α0 (57)

we understand 1, if γα0 = 1; if γα0 �= 1, then we understand limn

βα0+n =sup

nβα0+n of the following ω-sequence

Eα0,α0+1, E(α0,α0+2), ... E(α0,α0+n+1), ... (58)

where E(α0,α0+1) = γα0 and

E(α0,α0+n+1) = γγ..

.γ

γα0+nα0+n−1

α0+1α0 , (59)

for each 1 ≤ n < ω, is understood in the usual way: we descend, begin-

ning with γγα0+n

α0+n−1, γγ

γα0+nα0+n−1

α0+n−2 ,...,γγ..

.γα0+n

α0+2

α0+1 γγ..

.γα0+n

α0+2

α0+1 , ..., up to γγ

γ...γα0+n

α0+2α0+1

α0 =E(α0,α0+n+1).

Now, first of all, we shall prove the following lemmas.Lemma 2. Let ε0 and ε1 be the first and the second ε-numbers in

Cantor’s sense. Then for each γ, ε0 ≤ γ < ε1, we have

E(0,ω)(γ) = ε1; (60)

in particular,

E(0,ω)(ε0) = εεε..

.

00

0 = ε1. (61)

Moreover, for each ω-sequence of ordinals γ0, γ1, ..., γn, ... such that ε0 ≤γn < ε1, 0 ≤ n < ω, we have

E(0,ω){γ0, {γ1, {γ2, {...}}}}= γγ

γ...

21

0 = ε1. (62)

Proof. Since by the third equation in (52), γε10 = ε1, we have γγ1

0 <

γε10 = ε1 as well as γγ2

1 < γε11 = ε1, and hence γ

γγ21

0 < ε1. The same

argument says that γγ

γ...γn

21

0 < ε1, for each 0 ≤ n < ω. Consequently,

E(0,ω) = {γ0{γ1{γ1...}}} = γγ..

.

10 = lim

nE(0,n) = lim

nγ

γγ..

.γn

21

0 ≤ ε1. (63)


In particular, for γ = γn, 0 ≤ n < ω, E(0,ω)(γ) = γγ...

≤ ε1.In spite of the fact that E(0,ω)(γ) satisfies the equation (43) and seems

to be an ε-number, it is not a definition in Cantor’s sense, and it mightbe something different from Cantor’s classical ε-numbers and X(0,ω)(γ) =

γγ...

< ε1. Why not? We shall show now that this is not the case. Clearly,by ε0 ≤ γ, E(0,ω)(ε0) ≤ E(0,ω)(γ) and, by ε0 ≤ γn, 0 ≤ n < ω, E(0,ω)(ε0) ≤

E(0,ω) = γγ

γ...

21

0 . On the other hand, by ω < ε0, we obtain

E(0,n+1) = ωω...ω

εε00

< εε..

.εεε00

0

00 = X(0,n+1)(ε0). (64)

Since ε0 + 1 < εε00 < ε1, by (48) and [Theorem] C, we obtain

ε1 = limn

E(0,n+1) = limn

ωω...ω

εε00

≤ limn

εε..

.εεε00

0

00 = lim

nE(0,n+1)(ε0) ≤

≤ limn{γ0{γ1{γ2{...{γn}}}}} = γ

γγ..

.γn

21

0 ≤ ε1.

(65)

Consequently, E(0,ω) = γγ

γ...

21

0 = ε1, which completes the proof of Lemma 2.�

Lemma 3. Let ε′ and ε′′ be neighboring ε-numbers in Cantor’s sense.Then for each γ, ε′ ≤ γ < ε′′, we have

E(0,ω)(γ) = ε′′; (66)

in particular,

E(0,ω)(ε′) = ε′ε

′ε′...

= ε′′. (67)

Moreover, for each ω-sequence of ordinals γ0, γ1, ..., γn, ... such that ε′ ≤γn < ε′′, 0 ≤ n < ω, we have

E(0,ω){γ0, {γ1{γ2{...}}}} = γγ

γ...

21

0 = ε′′. (68)

The Proof is absolutely similar to the proof of Lemma 2.Lemma 4. Let ε0, ε1, ε2, ... be an ascending ω-sequence of ε-numbers.

Then E(0,ω) = εεε..

.

21

0 = limn

εn, and hence is an ε-number in Cantor’s sense.

58 JU. T. LISICA

Proof. Since ε0 < ε1 < ε2 < ... we have, by (52), E(0,n+1) = εεε..

.εn

21

0 =εn. Consequently, by Cantor’s Theorem E, E(0,ω) = lim

nE(0,n+1) = lim

nεn

is an ε-number. �Lemma 5. For any ordinal number γ ≥ ω, E(0,ω)(γ) = γγγ..

.

is anε-number in the sense of Cantor. Moreover, for each increasing ω-sequence

of ordinals γ0, γ1, ..., γn, ... such that ω ≤ γ0, E(0,ω) = γγ

γ...

21

0 is an ε-numberin Cantor’s sense.

Proof. By (45), we have ωγ ≥ γ and hence γ < E(γ). This is thewell-known fact that for each ordinal γ, there is an ε-number greater than γ([56], p. 327). Let ε′′ be the least of such ε-numbers. Take the preceeding ε′

which always exists because ε′′ cannot be a limit of ε-numbers, otherwise, ε′′

could not be the smallest ε-number greater than γ. Obviously, ε′ ≤ γ < ε′′.Then we apply Lemma 3 and obtain E(0,ω)(γ) = ε′′.

Let now γ0, γ1, ..., γn, ... be an increasing ω-sequence, i.e., γ0 ≤ γ1 ≤ ... ≤γn ≤ .... If it is stable, i.e., γn0 = γn0+1 = γn0+2 = ..., we put γ = γn0 and

apply the first assertion of Lemma 5, i.e. E(n0,ω) = γγ

γ...

n0+2n0+1

n0 = γγγ...

= ε′′.

Then, clearly, E(0,ω) = γγ

γ...γε′′n0−1

21

0 = ε′′.If it is not stable, then without loss of generality we can assume that

γ0 < γ1 < ... < γn < .... Suppose now that there are only finite ε-numbersbetween these ε-numbers, e.g., γ0 and γn0, and denote the greatest of themby ε′. Then, clearly, E(n0,ω) = ε′′ and E(0,ω) = ε′′ as in the previouscase. Finally, suppose that there is a ascending ω-sequence εν0 < εν1 < ...with is cofinal to a ascending ω-sequence γ0 < γ1 < .... Then, evidently,ε′′ = lim

nγn = lim

nεn and the proof that E(0,ω) = ε′′ is similar to that for

Lemma 4. �Remark 18. It is not true that for an arbitrary ω-sequence of ordinals

γ0, γ1, ..., γn, ... the limit power E(0,ω) = γγ..

.γ..

.

n

10 is an ε-number.

Indeed, consider the following ω-sequence ω, 2, 2, ... , then

ω22...

= limn

ω22...2

= ωlimn

22...2

= ωω . And ωω is not an ε-number.

Indeed, 2ωω= 2

limn

ωn

= limn

2ωn= lim

nωωn−1

= ωlimn

ωn−1

= ωωω> ωω. The

moral is that there are many ω-sequences of ordinals and correspondingskand-exponents whose limit-powers are equal to ε-numbers; on the other


hand, there are also a lot of ω-sequences of ordinals and the correspondingskand-exponents whose limit-powers are not ε-numbers at all.

Remark 18 allows us to give the followingDefinition 17. By an ε-number we understand any ordinal number of

the formε = E(0,ω)(γ), γ ≥ 1, (69)

and also for an arbitrary set E of such ε-numbers in (69) its supremum, i.e.,

ε′ = supε∈E

E . (70)

It is clear that all ε-numbers in Definition 17 are ε-numbers in the senseof Cantor except two numbers: 1 because 11 = 1 and (43) holds, and ω,because 2ω = ω and (43) holds, too. Let us denote these first two ε-numbersby ε0 and ε1, respectively. Since ε2 = E(0,ω)(ω), by Lemma 2, we obtainthat it is the least ε-number in the sense of Cantor; and all finite indexesof ε-numbers in the sense of Cantor are shifted by adding 2.

Definition 18. Let α = ωκ, κ ≥ 1. Then by a limit-power with thebasis γ > 1 and the same exponents we understand the skand-exponentE(0,α)(γ), given by the following transfinite recursion:

E(0,ω)(γ) = ε1, κ = 1; (71)

E(0,ω2)(γ) = E(0,ω)(ε1) = ε2, κ = 2; (72)..................................

E(0,ωn)(γ) = E(0,ω)(εn−1) = εn, κ = n; (73)..................................

E(0,ωω)(γ) = supn

E(0,ωn)(γ) = supn

E(0,ω)(εn−1) = εω, κ = ω; (74)..................................

In the general case

E(0,ωκ)(γ) = E(0,ω)(εκ−1) = εκ, κ − 1 < κ; (75)

E(0,ωκ)(γ) = supλ<κ

E(0,ωλ)(γ) = supλ<κ

E(0,ωλ)(γ) = εκ, � ∃ κ − 1. (76)

In accordance with Remark 18 and Definition 18, we are going to de-scribe in canonical form all ε-numbers.

Theorem 4. There is a one-to-one correspondence between all ordinalnumbers ωκ, 0 ≤ κ ∈ On and all ε-numbers εκ, defined in Definition 17,as follows: ε0 = E(0,1)(1) and εκ = E(0,ωκ)(2), κ ≥ 1.

Proof. If κ = 0, we put ε0 = 1. If κ > 0, putting in Definition 18γ = 2, εκ = εκ, we obtain, by Lemmas 2,3,4,5, a successive enumeration ofall ε-numbers in the sense of Definition 17 and thus all ε-numbers in thesense of Cantor. �

60 JU. T. LISICA

Theorem 4′. Put ε0 = 1 and ε1 = ω. Then for κ > 1 one hasthe following formulas for all ε-numbers: εκ+1 = E(0,ω)(εκ) and εκ+ω =lim

0≤n<ωE(0,ω)(εκ+n).

Proof is inside the the proof of Theorem 4.It is easy to see that if α �= ωκ, κ > 0, then E(0,α)(2) is not an ε-number.

Indeed, put α = ω + 1, then E(0,ω+1) = E(1,ω+1) = ω2; but 2ω2= ωω �= ω2.

Corollary 6. The set E(α) = {εκ| εκ < ωα}, i.e., the set of all ε-numbersεκ such that εκ less than the initial number ωα has a power greater thanor equal to ωα.

Proof. If the power of E(α) were less than the power of ωα, then ε =sup

εκ∈E(α)

εκ would be greater than all of the elements in E(α), and it would

be an ε-number whose power would be less than ωα, because the latter isnot the limit of a transfinite sequence of smaller powers. Thus E(α) ∪ {ε}would be larger than E(α), which is in contradiction with the maximalityof E(α). Consequently, |E(α)| ≥ |ωα| (|E(α)| and |ωα| mean the cardinalityof E(α) and ωα, respectively).

Corollary 7. The set E(α) = {εκ| εκ < ωα}, i.e., the set of all ε-numbers εκ such that εκ less than the initial number ωα is a well-orderedset of the ordinal type ωα and thus has the power of ωα.

Proof. For each κ < ωα, clearly, εκ < ωα. And this is a one-to-onecorrespondence between the set (0, ωα) = {κ| κ < ωα} and the set E(α).Moreover, if κ′ < κ′′ < ωα, then εκ′ < εκ′′ , i.e., the ordinal type (0, ωα)is the same as E(α). Since by Corollary 6, |E(α)| ≥ |ωα|, we obtain that|E(α)| = |ωα|.

Corollary 8. Every initial number ωα is an ε-number.Proof. Clearly, ωα = lim

εκ∈E(α)

εκ and thus, by Definition 17, it is an

ε-number.Remark 19. The assertion of Corollary 8 was given in [56] and proved

in the particular case ω1. It is well-known that each initial number ωα hasa form ωα = ωλ for some ordinal λ (see, e.g., [36], Chap. VIII, §3, Theorem9), but usually no one points out that λ = ωα.

Corollary 9. The well-ordered proper class of all ε-numbers is isomor-phic to On.

The Proof is similar to that for Corollary 7. �Remark 20. The limit power E(0,ωκ)(γ), κ ≥ 1, in Definition 18 should

be better called the quantified power, because actually we quantify expo-nents by ω-sequences of the same exponents: 1 < γ < ω, then ω, then ε2,ε3,..., εω ,..., just to simplify the algorithm of transfinite recursion. We coulddo the same thing by a continued exponentials process, e.g., E(0,ω1)(2):


E(0,2)(2) = 22,..., E(0,n)(2) = 22...2

,...E(0,ω)(2) = 22...

= ω = ε1;

E(0,ω+1)(2) = ω2, E(0,ω+2)(2) = ω22,... E(0,ω2)(2) = ω22..

.

= ωω = εε11 ;

E(0,ω2+1)(2) = ωω2, E(0,ω+2)(2) = ωω22

,... E(0,ω3)(2) = ωω22..

.

= ωωω,...,

E(0,ωn)(2) = ωω..

.ω

,...E(0,ω2)(2) = ωωωω...

= εεεε1..

.

11

1 = ε2;.................................E(0,ω2+1)(2) = ε2

2, E(0,ω2+2)(2) = ε22

2 ,..., E(0,ω2+ω)(2) = ε22...

1 = εω2 ,

E(0,ω2+ω+1)(2) = εω2

2 , E(0,ω2+ω+2)(2) = εω22

2 ,...E(0,ω22)(2) = εε22 ;

.................................

E(0,ω3)(2) = εεεε..

.

22

22 = ε3;

.................................

E(0,ωω)(2) = εεεε..

.

43

21 = εω;

..................................

E(0,ω1)(2) = εε..

.εε..

.ε..

.

α

ω+1ω

21 = 2E(1,ω1)(2) = 2ω1 = ω1, where 1 ≤ α < ω1.

There is a similar process in the case α = ωλ, λ ≥ 2, i.e., in the calcula-tion of E(0,ωλ).

Proposition 21. For an arbitrary ordinal number α > 0 and any1 < γ < ω, E(0,α)(γ) can be expressed by the following unique formula:

E(0,α)(γ) = γE(1,α)(γ) = εε..

.εε..

.εγ..

.γ

ηn

η2η1

η1η1 , (77)

where εη1 > εη2 > ... > εηn are ε-numbers, η1 > η2 > ... > ηn > 0 areordinal numbers, and the quantization of them, and of γ in the exponentsis given by β1, β2, ..., βn, βn+1, respectively, 0 ≤ βi < ω, i = 1, 2, ..., n, n+1.

Proof. By Cantor’s normal form of α, it may be represented uniquelyas

α = ωη1β1 + ωη2β2 + ... + ωηnβn + βn+1, (78)

where η1, η2, ..., ηn is a descending sequence of ordinal numbers > 0 andnatural numbers β1, β2, ..., βn, βn+1 are ≥ 0 (see, e.g., [36], Chap. VII, §7,Theorem 2). Then, by Theorem 4 and Remark 20, we obtain (77). �

62 JU. T. LISICA

Proposition 22. For every ordinal number γ ≥ 2, the hyper-skandE(0,Ω)(γ) is the greatest ε-class-number, i.e.,

E(0,Ω)(γ) = γE(1,Ω)(γ) = limα∈On

εα = εΩ = Ω = γΩ. (79)

Proof. It is clear that for each ordinal number α > 0, E(α,Ω)(γ) is equalto E(0,Ω)(γ). In other words, the skand E(0,Ω)(γ) is self-similar, becauseeach remainder (α, Ω) as an ordered class is equal to (0, Ω), i.e., (α, Ω) and(0, Ω) are isomorphic as ordered classes, in particular, for α = 1. Since, byCorollary 9, the class of all ε-numbers has of the same ordinal type as Ω

we obtain that limα∈On

εα = limα∈On

α = Ω and consequently, γΩ = γlim

α∈Onεα

=

limα∈On

γεα = limα∈On

εα = Ω, i.e., Ω satisfies (43), and thus Ω is an ε-class-

number. �

12. Applications to generalized real fractionsLikewise, we denote here a transfinite α-sequence xα0, xα0+1, ..., xα′, ...,

where α0 ≤ α′ < α < Ω and xα′ ∈ UΩ, as a special skand

X(α0,α) = {xα0, {xα0+1, {...{xα′, {...}}}}} (80)

as a descending sequence of two-elements sets X(α0,α) � X(α1,α) � ... �X(α′,α) � ..., where X(α′,α) = {xα′ , X(α′+1,α)}, α0 ≤ α′ < α.

Let α be an ordinal of the 2nd type, i.e. that having no predecessor;in particular, 0 is an ordinal of the 2nd type; its form α = ων, whereν ≥ 0, is known. Recall also that ordinals of the 1st type are those havingpredecessors. (We have already used other terminology above as a limitordinal number and an ordinal number which is not a limit number; anordinal of the 2nd type and of the 1st type are shorter, and we now preferthe latter terminology.)

Consider now for a fixed α = ων, ν ≥ 1, the set Aα of all special skands(henceforth, in short: skands) X(0,α) whose components Xα′ = 0 or Xα′ = 1and as above we do not differ individuals 0 and 1 with their singletons {0}and {1} as it demands the definition of Xα′: we simplify a notation withoutlosting the sense of matter.

We endow Aα with the following lexicographic linear ordering: X(0,α) <

Y(0,α) iff there is an α′, 0 ≤ α′ < α, such that Xα′ = 0 and Yα′ = 1 and atwhichever βth place, β < α′, the elements are equal; i.e., Xβ = Yβ . If inaddition Xβ = 1 and Yβ = 0, for all β > α′, then for such pairs only, thereare no skands Z(0,α) in Aα with X(0,α) < Z(0,α) < Y(0,α). We call thosepairs of neighboring skands twins. We shall identify them and denote theobtained new element in the canonical form, i.e., of a greater Y(0,α), not


forgetting that there is a different form of it, i.e., of a smaller X(0,α), andusing it when it is convenient.

Definition 18. By Rα|[0,1] = [0(0,α), 1(0,α)] we denote the quotient setAα/∼ of Aα (∼ identifies each pair of twins as one element) with the quo-tient linear ordering and call it a generalized real number (more precisely,fractional) unit interval of the power 2|α|.

Here 0(0,α) and 1(0,α) are minimal and maximal elements (integers) ofRα|[0,1], i.e., skands with 0 and 1 at all places, respectively.

Definition 19. By Qα|[0,1] we denote the subset of Rα|[0,1] of all skandsX(0,α) which are eventually 0 or 1.

In particular, we distinguish in Qα|[0,1] dyadic fractions, i.e. 12α′ as X(0,α)

such that Xα′−1 = 1 and Xβ = 0 for all β �= α′−1, for each ordinal numberα′ of the 1st kind, 1 ≤ α′ < α, (which is a twin to Y(0,α) with Yβ = 0 for0 ≤ β < α′, and Yβ = 1, for α′ ≤ β < α) and also 1

2α′ as X(0,α) such thatXβ = 0, for 0 ≤ β < α′, and Xβ = 1 for all β ≥ α′, for each ordinal numberα′ of the 2nd kind, 0 ≤ α′ < α. In other words, in short, 1

2α′ are skandsX(0,α) which are eventually 1.

Proposition 23. Rα|[0,1] and Qα|[0,1] are the dense linear orderings andQα|[0,1] is dense in Rα|[0,1].

The Proof is an immediate consequence of Definitions 18 and 19 to-gether with the definition of the ordering on Aα/∼.

Theorem 5. The space Rα|[0,1] is continuous: i.e., every non-emptysubset S of Rα|[0,1] has a smallest upper bound M(0,α) = sup S and a great-est lower bound m(0,α) = inf S in Rα|[0,1].

Proof. If there exists a maximal element max S in S, then sup S =max S, if there exists a minimal element min S in S, then inf S = min S.

Consider now the case when S has no maximal element and prove thatthere exists M(0,α) = sup S in Rα|[0,1], i.e., M(0,α) ∈ Rα|[0,1] such that forall X(0,α) ∈ S we have X(0,α) < M(0,α) and for each Y(0,α) ∈ Rα|[0,1] suchthat Y(0,α) < M(0,α) there is Z(0,α) ∈ S such that Y(0,α) < Z(0,α).

Indeed, there exists the smallest ordinal α1 ≥ 0 such that there is anelement X1

(0,α) ∈ S with X1α1

= 1 and for every X(0,α) ∈ S, Xβ = 0, for each0 = α0 ≤ β < α1 (if α1 = 0, then conditions Xβ = 0, β < α1, are absent).Otherwise, S should be {0(0,α)} or the empty set ∅, which is impossible byassumption.

We shall define M(0,α) ∈ Rα|[0,1] by induction on its non-trivial compo-nents, putting at the beginning Mα1 = 1 and Mβ = 0, for each 0 ≤ β < α1,and then define the following subset S1 = S \ {X(0,α)| X(0,α) ∈ S, X(0,α) <

X1(0,α)} of S = S0.

64 JU. T. LISICA

Since S has no maximal element and X1(0,α) ∈ S, there exists the smallest

ordinal α2 > α1 such that there is an element X2(0,α) > X1

(0,α) in S1 withX2

α2= 1, and for every X(0,α) ∈ S1, Xβ = 0, for each α1 < β < α2.

We continue to define M(0,α) for the next series of indexes by puttingMα2 = 1 and Mβ = 0, for each α1 < β < α2, and define now S2 =S1 \ {X(0,α)| X(0,α) ∈ S1, X(0,α) < X2

(0,α)}.Suppose that, for each 1 ≤ k ≤ n < ω, we have already found the

smallest ordinal αk > αk−1 and elements Xk(0,α) > Xk−1

(0,α)such that Xk

αk= 1

and in addition for every X(0,α) ∈ Sk−1 its components Xβ = 0, whereαk−1 < β < αk. Suppose also that we have already defined the nextcomponents of M(0,α), by putting Mαk

= 1 and Mβ = 0, for each αk−1 <

β < αk, as well as the set Sk = Sk−1 \ {X(0,α)| X(0,α) ∈ Sk−1, X(0,α) <

Xk(0,α)}. Notice that in this induction we put formally X0

(0,α) = 0(0,α).Since Sn has no maximal element and Xn

(0,α) ∈ S, there exists the small-est ordinal αn+1 > αn such that there is an element Xn+1

(0,α) > Xn(0,α) in Sn

with Xn+1αn+1

= 1 and for every X(0,α) ∈ Sn, Xβ = 0, for each αn < β < αn+1.We put Mαn+1 = 1 and Mβ = 0, for each αn < β < αn+1, and defineSn+1 = Sn \ {X(0,α)| X(0,α) ∈ Sn, X(0,α) < Xn+1

(0,α)}.

Thus Sn are defined for all 0 ≤ n < ω and we can consider their in-tersection

⋂n

Sn. If⋂n

Sn = ∅, i.e., there are no more elements X(0,α) in

S with Xβ = 1, β > αn, for each 0 ≤ n < ω, then we put Mβ = 0, forall αω ≤ β < α, where αω = lim

nαn. Since for each 0 ≤ α′ < α, Mα′ has

been already defined, we obtain an element M(0,α) ∈ Rα|[0,1] and we claimthat it is sup S. Indeed, by construction, for every X(0,α) ∈ S, we haveX(0,α) < M(0,α). If Y(0,α) ∈ Rα|[0,1] and Y(0,α) < M(0,α), then there is aminimal index αn, 0 ≤ n < ω, such that Yαn = 0 and Mαn = 1. TakeXn

(0,α) ∈ S. It is clear that Y(0,α) < Xn(0,α). Thus in this case the existence

of sup S is proved.If

⋂n

Sn �= ∅, then we define Sω =⋂n

Sn. Since Sω �= ∅ there exists the

smallest ordinal αω+1 ≥ αω such that there is an element Xω+1(0,α)

in Sω with

Xω+1(0,α) > Xn

(0,α), for each 0 ≤ n < ω, Xω+1αω+1

= 1, and for every X(0,α) ∈Sω, Xβ = 0, for each αω ≤ β < αω+1 (if αω+1 = αω, then conditionsXβ = 0, β < αω+1, are absent). We put Mαω+1 = 1 and Mβ = 0, for eachαω ≤ β < αω+1 and define the following set Sω+1 = Sω \ {X(0,α)| X(0,α) ∈Sω, X(0,α) < Xω+1

(0,α)}. Then we continue our algorithm as above.

Since each step of our inductive construction enlarges the index αν ,1 ≤ ν, at least by 1, we shall exhaust all of 0 ≤ α′ < α and obtain an


element M(0,α) ∈ Rα|[0,1] such that Xν(0,α) < M(0,α), for each ν ≥ 1. Since⋂

νSν = ∅ (otherwise, M(0,α) should be an elements of

⋂ν

Sν and thus the

greatest element of S) we conclude that M(0,α) = sup S. Indeed, for eachX(0,α) ∈ S, X(0,α) < M(0,α) and if Y(0,α) ∈ Rα|[0,1] and Y(0,α) < M(0,α), thenthere is the smallest ordinal αν condidered above such that Yαν = 0 andMαν = 1. Take Xν

(0,α) ∈ S and by construction Y(0,α) < X(0,α). �The proof of the existence of inf S in the case when S has no minimal

element is absolutely similar. We omit details, but there is another proof ofit. Putting S∗ = {1(0,α) − X(0,α)| X(0,α) ∈ S}, we obtain S∗ ⊂ Rα|[0,1] andS∗ has no maximal element. By the above proof, there exists a smallestupper bound M(0,α) of S∗. If we put m(0,α) = 1(0,α) − M(0,α), then it is agreatest lower bound of S. (For the meaning of 1(0,α) −X(0,α) see the nextparagraph.) �

Theorem 6. The covering dimension of a topological space Rα|[0,1] inthe order topology is equal to 1, i.e., dim Rα|[0,1] = 1.

Proof. It is well known that every linearly ordered space X is heredi-tarily normal ([11] Bourbaki, [1948]).

For every normal space X dim X ≤ Ind X ([57] Vedenissoff, [1939]).For every space Y the properties dim Y = 0 and Ind Y = 0 are equiv-

alent and have as their consequence the normality of Y ([2] Chap. II, §3,Proposition 3, p. 170).

It is also known that if every hereditarily normal space X is a union oftwo spaces Y and Z such that Ind Y = 0 and Ind Z = 0, then Ind X ≤ 1([35] Katetov, [1951]).

Putting Y = Qα|[0,1] and Z = Rα|[0,1] \ Y , we notice that since Y andZ are dense in themselves and in Rα|[0,1], we conclude that ind Y = 0and ind Z = 0; therefore, they are hereditarily disconnected ([30] Haus-dorff, [1914]), and being linearly ordered, are strongly zero-dimensional([31] Herrlich, [1965]), which is equivalent to Ind Y = 0 and Ind Z = 0 ([23]Engelking, [1977]). Then for Rα|[0,1] = Y ∪ Z we obtain Ind Rα|[0,1] ≤ 1.

Clearly, since Rα|[0,1] is continuous then ind Rα|[0,1] = 1 and hence,by 1 = ind Rα|[0,1] ≤ Ind Rα|[0,1] = 1 and dim Rα|[0,1] ≤ Ind Rα|[0,1], weobtain a desired equality dim Rα|[0,1] = 1. Otherwise, if dim Rα|[0,1] = 0,then Ind Rα|[0,1] = 0, which is false, because Ind Rα|[0,1] = 1. �

The nature of the 1-dimensional manifold Rα|[0,1] (a generalized realnumber unit interval) which comes from the real number unit intervalRω|[0,1] is illustrated by the following figures, where α = ω · 2. The firstfigure is a bifurcation of a rational number, for a example, x = 1

2 :

66 JU. T. LISICA

�

��

� �

0(0,ω) = (0, 000...) 1(0,ω) = (0, 111...)

0(0,ω2) = (0, 000...; 000...) 1(0,ω2) = (0, 111...; 111...)

0(0,ω2) 1(0,ω2)

��

��

��

��

��

��

��

��

12

0, 011...; 111... = 12 = 0, 100...; 000...

0, 011...; 000...

0, 011...; 000...

�= 0, 100...; 000...

0, 100...; 111...

0,011...= =0,100...

Fig. 2

and the second figure is a bifurcation of an irrational number, for example,1π :

�

��

� �

0(0,ω) 1(0,ω)

0(0,ω2) 1(0,ω2)

0(0,ω2) 1(0,ω2)

��

��

��

��

1π = 0, 01101...

0, 01101...; 000...

0, 01101...; 000...

�= 0, 01101...; 111...

0, 01101...; 111...

Fig. 3We can also demonstrate a canonical embedding iω·2ω : Rω|[0,1] → Rω·2|[0,1]

which preserves the linear ordering, given by iω·2ω (X(0,ω)) = Y(0,ω·2), whereYβ = Xβ, 0 ≤ β < ω, and Yβ = 0, ω ≤ β < ω · 2.

Moreover, for each α = ω · ν and β = ω · μ, where ν ≤ μ, there isa canonical embedding iβα : Rα|[0,1] → Rβ|[0,1] which preserves the linearordering, given by iβα(X0,α) = Y(0,β), where Yγ = Xγ , 0 ≤ γ < α, andYγ = 0, α ≤ γ < β.

We can also formally consider the case where α = Ω = ω · Ω and therational skands X(0,Ω), which are eventually 0 or 1, form a proper classQΩ|[0,1] in NBG−. As to RΩ|[0,1], i.e., the union of rational and irrationalskands X(0,Ω), it is not an object in NBG− because irrational skands are


elements of a super-class and outside of NBG−-type theory. Nevertheless,one can consider a canonical embedding iΩα : Rα|[0,1] → RΩ|[0,1] whichpreserves the linear ordering, given by iΩα (X(0,α)) = Y(0,Ω), where Yγ = Xγ ,0 ≤ γ < α, and Yγ = 0, α ≤ γ < Ω. Moreover, QΩ|[0,1] =

⋃α<Ω

iΩα (Rα|[0, 1]),

where α = ω · ν, 1 ≤ ν < Ω.One can also prove that QΩ|[0,1] and RΩ|[0,1] are dense in themselves;

QΩ|[0,1] is dense in RΩ|[0,1] and RΩ|[0,1] is continuous. It is also clear thatind QΩ|[0,1] = 0 and ind RΩ|[0,1] = 0; therefore, they are hereditarily dis-connected, and being linearly ordered, are strongly zero-dimensional, whichis equivalent to Ind Y = 0 and Ind Z = 0. Then for RΩ|[0,1] = Y ∪ Zwe obtain Ind RΩ|[0,1] ≤ 1. Clearly, since RΩ|[0,1] is continuous, thenind RΩ|[0,1] = 1 and hence, by 1 = ind RΩ|[0,1] ≤ Ind RΩ|[0,1] = 1 anddim RΩ|[0,1] ≤ Ind RΩ|[0,1], we obtain a desired equality dim RΩ|[0,1] = 1.Otherwise, if dim RΩ|[0,1] = 0, then Ind RΩ|[0,1] = 0, which is false, becauseInd RΩ|[0,1] = 1.

Unfortunately, all these natural arguments cannot be applied, becauseall references are to results valid only for topological spaces which are sets,and there are no similar results (even definitions of topology, dimensions,normality, etc.) for point proper classes, and it is not clear how to pass thisgap. We are in a situation where something is evident, but there are noresources to prove it. We can only formulate the following conjecture.

Conjecture 3. RΩ|[0,1] is a maximally dense one-dimensional contin-uum whose elements (mostly hyper-classes) are limits of QΩ|[0,1], i.e., limitsof Ω-sequences whose terms are sets. �

13. Additive and multiplicative operations on generalized frac-tions

In the next paragraph we shall formally extend for each ordinal α = ων,1 ≤ ν ≤ Ω, the closed unit Rα|[0,1] to the set (hyper-class in the caseν = Ω) Rα of generalized real numbers which is a one-dimensional linearlyordered continuous dense homogeneous point set (a dense homogeneouspoint manifold in the sense of Cantor). Unfortunately, only for ν = 1,i.e. α = ω, Rω is supplied with addition and multiplication which areassociative, commutative and distributive; moreover, Rω = R is the fieldof real numbers. If ν > 1, then it is not easy to completely define additionand multiplication in Rα, although it has been tried in [6]. We introducehere our version, which is different both in notations and in intention fromthat in [6].

For ordinal numbers there is a commutative and associative natural sumand a natural product in the sense of Hessenberg ([32], 591-594); i.e., if

68 JU. T. LISICA

ordinal numbers ξ and η are represented in the form of normal expansionξ = ωξ1n1 + ωξ2n2 + ... + ωξrnr and η = ωξ1m1 + ωξ2m2 + ... + ωξrmr,respectively, where ξ1 > ξ2 > ... > ξr are ordinal numbers, n1, n2, ..., nr

and m1, m2, ..., mr are integers ≥ 0, then by definition the natural sum isthe following ordinal number:

ξ ⊕ η = ωξ1(n1 + m1) + ωξ2(n2 + m2) + ... + ωξr(nr + mr). (81)

In order to define the natural product α � β of the ordinal numbers αand β one multiplies their normal expansions as if they were polynomials ofvariable ω; multiplying two powers of number ω, one forms the natural sumof the exponents and arranges the terms obtained from the multiplicationaccording to decreasing exponents.

(Notice that instead of ω we can consider any ordinal number γ > 1 asa basis of normal expansion of ξ with 0 ≤ ni < γ, 1 ≤ i ≤ r, see [36], Chap.XII, §7, e.g., γ = 2; we use shall it below.)

Thus, the problem of the existence of the desired algebraic operationson Rα concerns only generalized real fractions of Rα|[0,1].

So addition “+”, subtraction “−”, multiplication “·” and division “/” arenot defined for all generalized real fractions X(0,α) and Y(0,α) in Rα|[0,1] butare defined for some of them. Here are typical cases of such a possibility:

1). For every X(0,α), Y(0,α) ∈ Rα|[0,1], such that Xα′ ≤ Yα′ , 0 ≤ α′ < α,we put Y(0,α) − X(0,α) = Z(0,α), where Zα′ = Yα′ − Xα′, 0 ≤ α′ < α. E.g.,X(0,α)−X(0,α) = 0(0,α) or, for each X(0,α) ∈ Rα|[0,1], 1[0,1]−X(0,α) is alwayswell defined.

If Y(0,α) = 12α′ and X(0,α) = 1

2α′′ , 0 ≤ α′ < α′′ < α, the subtraction1

2α′ − 12α′′ is well defined. Indeed, in the case when α′, α′′ are ordinal numbers

of the 2nd kind 12α′ − 1

2α′′ = Z(0,α), where Zβ = 1, for α′ ≤ β < α′′, andZβ = 0, for β < α′ or β ≥ α′′; in the case when α′, α′′ are ordinal numbersof the 1st kind 1

2α′ − 12α′′ = Z(0,α), where Zβ = 1, for α′ < β ≤ α′′, and

Zβ = 0, for β ≤ α′ or β > α′′; and at last in the case when α′ is an ordinalof the 1st kind and α′′ is an ordinal of the 2nd kind 1

2α′ − 12α′′ = Z(0,α),

where Zβ = 1, for α′ < β < α′′, and Zβ = 0, for β ≤ α′ or β ≥ α′′. (Notethat we use another notation in an appropriate case for twins.)

We will say that the interval [X(0,α), Y(0,α)] is of length 12α′ , if Y(0,α) −

X(0,α) = 12α′ , 0 ≤ α′ < α.

2). We can define X(0,α) + Y(0,α) in the case where, for each 0 ≤ α′ < α,Xα′ and Yα′ are both 0, or one of them is 1 and another one is 0. Then theresult is Z(0,α) = X(0,α) + Y(0,α) such that Zα′ = Xα′ + Yα′ , 0 ≤ α′ < α. Inparticular, if Z(0,α) = Y(0,α)−X(0,α), then X(0,α)+Z(0,α) = Z(0,α)+X(0,α) =Y(0,α).


If α′ is an ordinal of the 1st kind, then 12α′ + 1

2α′ = 12α′−1 , 0 ≤ α′ < α;

and if 1 ≤ α′ < α′′ < α, are arbitrary ordinal numbers of the 1st kind, then1

2α′ + 12α′′ = 1

2α′′ + 12α′ is Z(0,α) such that Zβ = 1, for β = α′, α′′; otherwise

Zβ = 0. If 0 ≤ α′′ < α is an ordinal of the 2nd kind and 0 ≤ α′ < α′′ < α

is an ordinal of the 1st kind, then 12α′ + 1

2α′′ = 12α′′ + 1

2α′ = Z(0,α) such thatZβ = 1, for β = α′ and β ≥ α′′; otherwise, Zβ = 0.

On the other hand, there are no magnitudes in Rα|[0,1] such as 12α′ + 1

2α′ ,for every 0 ≤ α′ < α which is an ordinal number of the 2nd kind, becausethere is no ordinal number α′ − 1; as well as 1

2α′ + 12α′′ , if α′ is an ordinal

number of the 2nd kind and α′ < α′′.Moreover, we can define addition X(0,α) + Y(0,α) in all cases which avoid

the addition of components of the latter cases when sums of dyadic fractionsdo not exist.

3). For 1 ≤ α′ < α, where α′ is an ordinal number of the 1st kind,∑α′≤β<α′+ω

12β = 1

2α′−1− 1

2α′+ω, where by an infinite sum we understand

the supremum of finite sums, if of course they are well defined. Indeed,1

2α′ + 12α′+1

+ ... + 12α′+n

+ ... = supn

( 12α′ + 1

2α′+1+ ...+ 1

2α′+n) = sup

n[( 1

2α′−1−

12α′ )+ ( 1

2α′+1− 1

2α′ )+ ...+( 12α′+n−1

− 12α′+n

)] = supn

( 12α′−1

− 12α′+n

) = 12α′−1

−infn

12α′+n

= 12α′−1

− 12α′+ω

.

4). We also put 12α′′ · 1

2α′ = 12α′′ L

α′ , 0 ≤ α′, α′′ < α.5). Each X(0,α) ∈ Rα|[0,1] can be divided by 2. Indeed, it is an immediate

consequence of the following propositions.Theorem 7 Each X(0,α) ∈ Rα|[0,1], in particular, 0 = 0(0,α) and 1 =

1(0,α) with 0 and 1 in all places, respectively, can be represented by thefollowing formula:

X(0,α) =∑

0≤α′<α

F (α′)2α′+1

, (82)

where the summation is taken over all ordinals 0 ≤ α′ < α, and F (α′)is equal to the element of the component Xα′ of X(0,α). In particular, for0 ≤ α0 < α, we obtain the following formula:

12α0

=∑

α0≤α′<α

12α′+1

=1

2α0+1+

12α0+2

+ ...+1

2ω+1+ ...+

12α′+1

+ ... (83)

The Proof is an immediate consequence of the above definitions.Thus, we can divide each X(0,α) ∈ Rα|[0,1] by 2, changing each term 1

2β

in (82) by 12β+1 and adding summands 1

2ωη+1 , if Xβ = 1, α′ ≤ β < α′ + ων,for some 0 ≤ α′ < α, where α′, is an ordinal number of the 2nd kind such

70 JU. T. LISICA

that α′ +ων < α, 1 ≤ η < ν, and the sum of all changed terms is the resultZ(0,α) = X(0,α)/2 = X(0,α) · 1

2 . �Lemma 6. Each interval [X(0,α), Y(0,α)] of length 1

2α′ , 0 ≤ α′ < α, canbe halved.

Proof. In fact, Y(0,α) − X(0,α) = 12α′ or Y(0,α) = X(0,α) + 1

2α′ = X(0,α) +1

2α′+1 + 12α′+1 . Consequently, Y(0,α) − 1

2α′+1 = X(0,α) + 12α′+1 and hence

[X(0,α), X(0,α)+ 12α′ ]∪[Y(0,α)− 1

2α′ , Y(0,α)] = [X(0,α), Y(0,α)]. Moreover, lengthsof [X(0,α), X(0,α)+ 1

2α′+1

] and [Y(0,α)− 12α′+1 , Y(0,α)] are equal to 1

2α′+1 = 12α′ · 12 .

�Although there are no magnitudes in Rα|[0,1] like n

2α′ , for each 2 ≤ n <

ω, where 0 ≤ α′ < α is an ordinal number of the 2nd kind, there aremagnitudes in Rα|[0,1] of multiplications 1

2β · 2α′for some ordinals 1 ≤

β, α′ < α = ων, ν ≥ 1, e.g., 12ω · 2ω = 1(0,α), which is really unexpected.

Proposition 24. For each α′, α′′ ∈ On, 0 ≤ α′, α′′ < α = ων, ν ≥ 1,the following formula

12α′ =

12α′⊕ω′′ · 2α′′

(84)

holds; in particular, 1ω · ω = 1

2ω · 2ω = 1(0,α).

Proof. Since 12α′ · 1

2α′′def= 1

2α′⊕ω′′ it is sufficient to show that 12α′′ ·2α′′

= 1.By Proposition 24, we obtain the following identities:

1 = 120 = 1

2 + 122 + 1

23 + ... = ( 122 + 1

23 + ...) + ( 122 + 1

23 + ...) =( 122 + 1

23 + ...) · 2 = 12 · 2 = [( 1

23 + 124 + ...) + ( 1

23 + 124 + ...)] · 2 =

( 123 + 1

24 + ...) · 22 = 122 · 22 = ... = [( 1

2n+1 + 12n+2 + ...)+

( 12n+1 + 1

2n+2 + ...)] · 2n−1 = ( 12n+1 + 1

2n+2 + ...) · 2n−1+( 12n+1 + 1

2n+2 + ...) · 2n−1 = ( 12n+1 + 1

2n+2 + ...) · 2n = 12n · 2n =

( 12n − 1

2ω + 12ω+1 + 1

2ω+2 + ...) · 2n = ... == ( 1

2ω − 12ω + 1

2ω+1 + 12ω+2 + ...) · 2ω =

( 12ω+1 + 1

2ω+2 + ...) · 2ω = 12ω · 2ω = ... =

= ( 12α′′+1

+ 12α′′+2

+ ...) · 2α′′= 1

2α′′ · 2α′′.

(85)

We also need the following lemma.Lemma 7. For every X(0,α) < Y(0,α) in Rα|[0,1] there are X ′

(0,α) < Y ′(0,α)

in Qα|[0,1] such that X(0,α) < X ′(0,α), Y ′

(0,α) < Y(0,α) and Y ′(0,α)−X ′

(0,α) = 12α′

for some 2 ≤ α′ < α; in particular, for each Y(0,α) there is 12α′ such that

12α′ < Y(0,α).

Proof. Since X(0,α) < Y(0,α) there exists α′, 0 ≤ α′ < α such thatXα′ = 0, Yα′ = 1 and Xβ = Yβ , for all 0 ≤ β < α′. There is a minimalα′′ > α′ such that Xα′′ = 0; otherwise, Xβ = 1, for all β > α′ and hence


Xα′ should be equal to 1. Consider X ′(0,α) such that X ′

α′′ = 1, X ′β = Xβ,

0 ≤ β < α′′, and X ′β = 0, β > α′′. Clearly, X(0,α) < X ′

(0,α) < Y(0,α).Consider Y ′

(0,α) such that Y ′α′′+1 = 1 and Y ′

β = X ′β, 0 ≤ β ≤ α′′. Clearly,

X ′(0,α) < Y ′

(0,α) < Y(0,α) and Y ′(0,α) − X ′

(0,α) = 12α′′+1

, i.e., α′ = α′′ + 1. �Definition 20. Let X(0,α) and Y(0,α) be two elements of Rα|[0,1] such

that X(0,α) ≤ Y(0,α) and Y(0,α) − X(0,α) is defined. Then the generalizedreal number l(0,α) = Y(0,α) − X(0,α) is called a length of the closed interval[X(0,α), Y(0,α)] and of the open interval (X(0,α), Y(0,α)).

Theorem 8. Let [X(0,α), Y(0,α)] ⊃ [X1(0,α), Y

1(0,α)] ⊃ ... ⊃ [Xα′

(0,α), Yα′(0,α)] ⊃

... be a system of embedded closed intervals of Rα|[0,1] such that infα′

lα′

(0,α) =

0(0,α); then there exists a unique element Z(0,α) ∈ Rα|[0,1] such that it be-longs to all these intervals, i.e.,

⋂α′

[Xα′(0,α), Y

α′(0,α)] = Z(0,α).

Proof. By Theorem 5, there exist M(0,α) = sup0≤α′<α

{Xα′(0,α)} and m(0,α) =

inf0≤α′<α

{Y α′(0,α)} in Rα|[0,1]. Then Z(0,α) = M(0,α) = m(0,α). Otherwise, by

Lemma 7, for m(0,α) < M(0,α) in Rα|[0,1] there are X ′(0,α) < Y ′

(0,α) in Qα|[0,1]

such that m(0,α) < X ′(0,α) < Y ′

(0,α) < M(0,α) and Y ′(0,α) − X ′

(0,α) = 12α′

for some 2 ≤ α′ < α, which is in contradiction with the assumption thatinfα′ lα

′(0,α) = 0(0,α), because [X ′

(0,α), Y′(0,α)] ⊂ [Xα′

(0,α), Yα′(0,α)], for each 0 ≤ α′ <

α. �Theorem 9. Rα|[0,1] in the order topology is a compact Hausdorff space.Proof. We already know that the linearly ordered space Rα|[0,1] is

normal. Let now γ = {Uλ} | λ ∈ Λ be an arbitrary covering of Rα|[0,1],consisting of open intervals Uλ of Rα|[0,1]. We have to prove that there ex-ists a finite subcovering of γ, which covers Rα|[0,1]. Suppose the contrary,and we cannot choose such a finite subcovering. By Lemma 6, we can halveRα|[0,1] = [X(0,α), Y(0,α)] (evidently, X(0,α) = 0(0,α) and Y(0,α) = 1(0,α)) andchoose one [X1

(0,α), Y1(0,α)] of the parts that cannot be covered by finite ele-

ments of γ. Then we halve [X1(0,α), Y

1(0,α)] and choose one [X2

(0,α), Y2(0,α)] of

the parts that cannot be covered by finite elements of γ. We continue thisprocess and conclude that [Xω

(0,α), Yω(0,α)] cannot be also covered by finite

elements of γ, otherwise, [Xn(0,α), Y

n(0,α)] can be covered by finite elements

of γ, which is in contradiction with our choice. We halve [Xω(0,α), Y

ω(0,α)]

and continue our choice for each 0 ≤ α′ < α. We have gotten a system[X(0,α), Y(0,α)] ⊃ [X1

(0,α), Y1(0,α)] ⊃ ... ⊃ [Xα′

(0,α), Yα′(0,α)] ⊃ ... of embedded

closed intervals of Rα|[0,1] such that infα′ lα

′(0,α) = 0(0,α). Then, by Theorem

72 JU. T. LISICA

8, there exists a unique element Z(0,α) ∈ Rα|[0,1] such that it belongs toall these intervals, i.e.,

⋂α′

[Xα′(0,α), Y

α′(0,α)] = Z(0,α). Since γ is a covering

of Rα|[0,1] there exists an element Uλ ∈ γ such that Z(0,α) ∈ Uλ. Sinceinfα′

lα′

(0,α) = 0(0,α) we conclude that there exists an ordinal number α′ < α

such that [Xα′(0,α), Y

α′(0,α)] ⊂ Uλ and one-element subcovering Uλ of γ covers

[Xα′(0,α), Y

α′(0,α)], which is in contradiction with our choice. Thus the assump-

tion that there is no finite subcovering of γ which covers Rα|[0,1] is wrong.�

14. Generalized real numbers and generalized straight linesNow we are going to extend Rα|[0,1], α = ων, ν ≥ 1, to the set Rα of all

generalized real numbers.By R+

α we denote Rα|[0,1]∪(Rα|(0,1])∗, where (L)∗ is the backwards linearordering of L = Rα|(0,1] = Rα|[0,1]\{0(1,α)

}, and identify 1(0,α) and (1(0,α))∗

with the ordinal 1, and each ordinal 2α′, 0 < α′ < α with ( 1

2α′ )∗ ∈ L∗ withthe obvious ordering, adding to the following already defined relations:Y < Z for each Y �= 1(1,α) in L and each Z �= (1(0,α))∗ in (L)∗. PuttingR−

α = (R+α )∗ and denoting (X(0,α))∗ by −X(0,α), for every X(0,α) ∈ R+

α ,identifying 0(1,α) and −0(1,α), we define Rα = R−

α ∪ R+α as generalized

real numbers with the obvious ordering, adding to the following alreadydefined relations: X(0,α) < Y(0,α), for each X(0,α) �= −0(1,α) in R−

α and eachY(0,α) �= 0(1,α) in R+

α . It is clear that Rα is a set of the power 2|α|.Similarly, we can extend Qα|[0,1] to the dense subordering subset Qα of

Rα and call it generalized rational numbers. Its cardinality is∑

α′<α

2|α′|.

One can easily prove that Qα is dense in Rα; dim Qα = 0, dim (Rα \Qα) = 0, dim Rα = 1; Rα is continuous, i.e., for every bounded setXα ⊂ Rα, there exists an interval [α0, α1] such that Xα ⊆ [α0, α1] hasa smallest upper bound and a greatest lower bound; every closed boundedset is compact; each Dedekind section in Rα has no gap.

In the case α = ωκ, κ ≥ 1 we can represent Rα in a more natural form.Definition 21. By a set R+

α of all non-negative generalized numbers weunderstand an extended system of embedded curly braces

...{−α′...{−1{0{1...{α′...}}}}} (86)

filled by 0 or 1 such that for each X(−α,α) ∈ R+α , Xα′ = 1 only for finite

number indexes, −α < α′ < 0. We consider on R+α the lexicographic

ordering identifying twins as above. If all non-negative places are filled by0 we have the usual ordinal numbers in Cantor’s normal form with base 2


and the lexicographic ordering of them which coincides with the usual one.By a set R−

α of all non-positive generalized numbers we understand thebackwards linear ordering (R+

α )∗, and denote X∗(−α,α) ∈ (R+

α )∗ by −X(−α,α).At last, by a set Rα of all generalized numbers we understand R−

α ∪R+α with

the natural identification 0+(−α,α) and 0−(−α,α) and a clear linear ordering.

Further we denote by X(−α,α) an arbitrary element of Rα which can bepositive, negative or zero.

Theorem 10. If α = ωκ, κ ≥ 1, then Rα is topologically and orderisomorphic to Rα in the sense of Definition 22.

Proof. We give only a sketch of a proof. It is enough to prove thatR+

α \ Rα|[0,1) is isomorphic to R+α \ Rα|[0,1) in the sense of Definition 22,

because Rα|[0,1) are isomorphic in both senses: notice that Rα|(0,1] are alsoisomorphic in both senses.

Since Rα|(0,1] and (Rα|(0,1])∗ are evidently topologically isomorphic it isenough to show that (Rα|(0,1])∗ and {X(−α,α) ∈ Rα | X(−α,α) ≥ 1(−α,α)} orRα|(0,1] and {X(−α,α) ∈ Rα | X(−α,α) ≥ 1(−α,α)} are topologically isomor-phic, respectively.

This isomorphism can be defined by the following transfinite induction:the first step is to show that Rα|[ 1

2,1] is isomorphic to Rα|[1,2]. It can be done

by putting in correspondence 12 to 2 and 1 to 1, respectively (we obviously

simplify the notation). By halving the intervals [12 , 1] and [1, 2] we puttheir centers in correspondence to each other, i.e., 1

2 + 122 to 1 + 1

2 , anddo the same (i.e., halving the intervals) with each corresponding interval[ 12 , 1

2 + 122 ] and [1+ 1

2 , 2] as well as [ 12 + 122 , 1] and [1, 1+ 1

2 ], respectively. Ofcourse, the limit ends will be in this natural correspondence and we continuehalving further and further, i.e., α times. It is clear that the closures ofthe corresponding isomorphic sets of halving are also isomorphic and theycoincide with Rα|[ 1

2,1] and Rα|[1,2], respectively.

We can do the same with Rα|[ 122

,1] and Rα|[1,22], respectively, and no-

tice that a new isomorphism restricted on Rα|[ 12,1] and Rα|[1,2], respec-

tively, will coincide with the previous one. The limit isomorphism betweenRα|( 1

2ω ,1] and Rα|[1,2ω) is obvious, which we extend to the isomorphismbetween Rα|[ 1

2ω ,1] and Rα|[1,2ω].In the same manner we show that Rα|[ 1

2ω+n , 12ω ] and Rα|[2ω,2ω+n], 1 ≤ n <

ω, are isomorphic. (Only notice here that if α �= ωκ, κ ≥ 1, then this stepwould be wrong.) And extend it to the isomorphism between Rα|[ 1

2ω2 , 12ω ]

and Rα|[2ω,2ω2] and hence between Rα|[ 12ω2 ,1] and Rα|[1,2ω2]. The further

74 JU. T. LISICA

steps are similar. We omit the details. The resultant ordering isomorphismbetween Rα in both senses will be an isomorphism, too. �

If each element of R(−α,α), α = ωξ, ξ ≥ 0, is considered as a geometricpoint, then we denote this point set by L(−α,α) and call it a generalizedstraight line. If ξ = 0, then we obtain a classical Euclidean line, whichas we know is uniquely defined by Hilbert’s axioms. Moreover, by one ofEuclid’s definitions: “The straight line is a line such that it is uniformlyarranged towards all its points”. It is not so in the cases where ξ > 0;i.e. there are different kinds of points, e.g., ξ = 1 and 1

2ω and 12ω+1 in

R(−ω1,ω1), which have a different structure to the right of them, though thesame structure to the left. We are going to state the following conjecture.

Conjecture 4. There exists a system of axioms including a general-ized Archimedian axiom, a generalized Cantor’s axiom of continuity, whichuniquely defines the generalized straight line L(−ωξ,ωξ), ξ ≥ 1, with an iso-morphism ϕ : L(−ωξ,ωξ) → R(−ωξ,ωξ).

In favour of this conjecture says the following corollary of Proposition26:

Corollary 10. For each generalized dyadic fraction a = 12α′ , α′ ∈ On,

1 ≤ α′ < α = ωξ, ξ ≥ 1, and arbitrary positive generalized real numberb ∈ R+

(−ωξ,ωξ)there exists an ordinal number ν, 1 ≤ ν < ωξ, such that

a · ν > b.Proof. Indeed, there exists an ordinal number β, 0 ≤ β < α = ωξ, such

that 2β > [b], where [b] is an integral part of b. Putting α′′ = α′ ⊕ β, weobtain ν = 2α′⊕β which, by Proposition 24, satisfies the desired condition,i.e., a · ν = 1

2α′ · ν = 12α′ · 2α′⊕β = 1

2α′ · 2α′ � 2β = ( 12α′ · 2α′

) · 2β = 1 · 2β > b.�

Note also that the geometry of such straight lines is different from theclassical one; i.e., L(−ω,ω). E.g., in the generalized plane, i.e., L2

(−ωξ,ωξ) =L(−ωξ,ωξ)×L(−ωξ ,ωξ), ξ ≥ 1, it is not a case that each of two different pointsof R2

(−ωξ,ωξ) belongs to a generalized line in R2(−ωξ,ωξ). For example, there

are straight lines in R2(−ωξ,ωξ) such as y = x, or y = −x, or x = const, or

y = const, but there is no line given by the following equation: y = 2x.So for possible straight lines in R2

(−ωξ,ωξ), ξ ≥ 1, a generalized version ofZeno’s paradox arises; however in the direction y = 2x, Zeno’s arrow doesnot even exist; it is totally destroyed. �

15. Elements of a generalized calculusDefinition 22. By an α′-sequence in Rα, ω ≤ α′ ≤ α ≤ Ω, α =

ων, α′ = ων′, 1 ≤ ν, ν′ ≤ Ω, we understand a generalized skand S(0,α′)whose components Sβ′ are elements of Rα, 0 ≤ β′ < α′. We denote it


temporarily by Sβ′, 0 ≤ β′ < α′; in the usual way its denotation is morecomplicated, i.e., {Sβ′}|β′<α′ and is called a transfinite sequence of typeα′ (see the corresponding definitions of course for transfinite sequences ofordinal numbers [52], p. 287).

Remark 21. So, for α > ω, there are many converging α′-sequences inRα, ω ≤ α′ ≤ α, where α′ = ων, ν ≥ 1. Indeed, since each generalized realnumber X(−α,α) ∈ Rα is the intersection of some α′-sequence of embeddedclosed intervals [X(−α,α), B

β(−α,α)] ⊃ [X(−α,α), B

β+1(−α,α)], 0 ≤ β < α′, (a

cofinal system of closed neighborhoods on the right at X(−α,α)) and is atthe same time the intersection of some α′′-sequence of embedded closedintervals [Aβ

(−α,α), X(−α,α)] ⊃ [Aβ+1(−α,α), X(−α,α)], 0 ≤ β < α′′, ω ≤ α′′ ≤

α, (a cofinal system of closed neighborhoods on the left at X(−α,α)) thenBβ

(−α,α) and Aβ(−α,α) are such α′- and α′′-sequences, respectively. Notice

that there are elements X(−α,α) ∈ Rα for which α′ = α′′ = α (generalizedintegers, generalized irrationals, some generalized rationals), but there areelements X(−α,α) ∈ Rα for which α′ < α or α′′ < α though not at the sametime; i.e., if α′ < α, then α′′ = α or if α′′ < α, then α′ = α(e.g., 1

2ω , 1− 12ω ,

etc.)Definition 23. A generalized real number X(−α,α) ∈ Rα is a limit of

α′-sequence Sβ, 0 ≤ β < α′ ≤ α, notation X(−α,α) = limβ→α′

Sβ , if for each

open interval (A(−α,α), B(−α,α)), A−(α,α) < B(−α,α), which contains X(−α,α),there exists an ordinal 0 ≤ β0 < α′ such that Sβ ∈ (A(−α,α), B(−α,α)), forall β0 < β < α′.

In this case a α′-sequence Sβ , 0 ≤ β < α′ ≤ α, is called convergent andX(−α,α) ∈ Rα is its limit. Clearly, if the α′-sequence Sβ, 0 ≤ β < α′ ≤ α,converges to X(−α,α) ∈ Rα, then this limit is unique.

Using the classical arguments we can easily prove the following theorems.Theorem 11. A mapping f : Rα → Rα is continuous in the ordering

topology if and only if for each element X(−α,α) ∈ Rα and every α′-sequenceSβ, 0 ≤ β < α′ ≤ α, such that lim

β→α′ Sβ = X(−α,α), then limβ→α′ f(Sβ) =

f(X(−α,α)).Proof. Let f : Rα → Rα be a continuous mapping and X(−α,α) an

arbitrary element in Rα. Consider any open interval (A′(−α,α), B

′(−α,α))

which contains f(X(−α,α)) and find an open interval (A(−α,α), B(−α,α))which contains X(−α,α) such that f(A(−α,α), B(−α,α)) ⊆ (A′

(−α,α), B′(−α,α)).

Since limβ

Sβ = X(−α,α) there exists an ordinal number β0 such that Sβ ∈

76 JU. T. LISICA

(A(−α,α), B(−α,α)), for all β0 < β < α′ ≤ α. Then evidently f(Sβ) ∈(A′

(−α,α), B′(−α,α)), for all β0 < β < α′ ≤ α.

Conversely, suppose the opposite, i.e., for every α′-sequence Sβ, 0 ≤ β <α′ ≤ α, such that lim

β→α′ Sβ = X(−α,α) we have limβ→α′ f(Sβ) = f(X(−α,α));

however f is not continuous at some point X(−α,α) in R(−α,α). Evidently, fis not continuous on the right at X(−α,α) or on the left at X(−α,α). We canconsider the first case; the second is similar. Thus, if f is not continuouson the right at X(−α,α), then there is an open interval (A′

(−α,α), B′(−α,α))

which contains f(X(−α,α)) such that for every embedded closed interval[X(−α,α), B

β(−α,α)

], 0 ≤ β < α′ ≤ α, which is cofinal in the system of

all neighborhoods on the right at X(−α,α), we have f([X(−α,α), Bβ(−α,α)

])

is not a subset of (A′(−α,α), B

′(−α,α)). Choosing in each [X(−α,α), B

β(−α,α)]

an element Sβ, such that f(Sβ) /∈ (A′(−α,α), B

′(−α,α)), 0 ≤ β < α′ ≤ α,

we see that limβ→α′ Sβ = X(−α,α) but lim

β→α′ f(Sβ) �= f(X(−α,α)), which is in

contradiction with our assumption; i.e., if for an α′-sequence Sβ we havelim

β→α′ Sβ = X(−α,α), then limβ

f(Sβ) = f(X(−α,α)). �Theorem 12. Let X(−α,α) and Y(−α,α) be elements of Rα. If

f : [X(−α,α), Y(−α,α)] → Rα (87)

be a continuous mapping such that f(X(−α,α)) and f(Y(−α,α)) have differentsigns, i.e., − and + or + and −, respectively, then there exists an elementZ(−α,α) in Rα such that f(Z(−α,α)) = 0(−α,α).

Proof. If in the interval [X(−α,α), Y(−α,α)] there are no integer ordinalnumbers except X(−α,α) and Y(−α,α), then the proof is classical here andis given by the method of our generalized dichotomy, or halving the inter-vals of the unit one. Comp., [20], Theorem 25, p. 41. If there are suchintegers, then choose one of them; let it be Z(−α,α) such that X(−α,α) <Z(−α,α) < Y(−α,α). Clearly, f(X(−α,α)) and f(Z(−α,α)) or f(Z(−α,α)) andf(Y(−α,α)) have different signs, and we choose that interval and denote itby [X1

(−α,α), Y1(−α,α)]. We continue this transfinite process up to the first

case where there are no integer ordinal numbers except Xβ(−α,α) and Y β

(−α,α),1 ≤ β < α; then the proof is also classical and is given by the method ofour generalized dichotomy or halving intervals of the unit one. Since weobtain a system of embedded closed intervals whose lengths converge to0(−α,α), by Theorem 8, there exists a unique element Z(0,α) ∈ Rα such thatit belongs to all these intervals. By the continuous property of f , f(Z(0,α))


has at the same time different signs, which is possible only in the case wheref(Z(0,α)) = 0(−α,α). �

One can consider other classical theorems of Mathematical Analysis inthe case of Rα since for ξ = 1 we have the usual real numbers R = Rω.

Theorem 13. Every closed interval [A(−α,α), B(−α,α)] of Rα is compact.Proof. Without loss of generality we can assume that A(−α,α) and

B(−α,α) are positive or negative ordinals and A(−α,α) < B(−α,α). More-over, we can restrict the proof to the case where [A(−α,α), B(−α,α)] ⊂ R+

α .Assume now that γ is a covering of [A(−α,α), B(−α,α)] ⊂ R+

α consisting ofopen sets in Rα which has no finite subcovering. Then we divide this in-terval by the point A(−α,α) +1, i.e., [A(−α,α), B(−α,α)] = [A(−α,α), A(−α,α) +1] ∪ [A(−α,α) + 1, B(−α,α)], the next [A(−α,α) + 1, B(−α,α)] by A(−α,α) + 2,etc. By Theorem 9, restrictions of γ on [A(−α,α), A(−α,α) + 1], [A(−α,α) +1, A(−α,α) + 2],...,[A(−α,α) + n, A(−α,α) + n + 1],... have finite subcoverings;we conclude that the restrictions of γ on each [A(−α,α)+n, B(−α,α)] have nofinite subcoverings, 0 ≤ n < ω. Hence [A(−α,α) + ω, B(−α,α)] has no finitesubcovering of the restriction of γ, if of course A(−α,α) + ω < B(−α,α). IfA(−α,α) + ω = B(−α,α), then B(−α,α) is covered by one element of γ andconsequently, [A(−α,α) +n, B(−α,α)] covers by this element for some naturaln. Contradiction. So we continue this transfinite process, passing all pos-sible limit ordinals and B(−α,α) with the same argument. If B(−α,α) is nota limit ordinal, then the impossibility of choosing a finite subcovering fromthe restriction of γ on [B(−α,α) − 1, B(−α,α)] contradicts Theorem 9. �

Theorem 14. A subspace X of Rα is a compact Hausdorff space if andonly if it is a bounded closed subset of Rα.

The Proof is classical, except with reference to Theorem 13.Theorem 15. Every continuous image of a compact Hausdorff space is

a compact Hausdorff space.The Proof is classical.Theorem 16. Every bounded subset S of Rα has a smallest upper bound

M(0,α) = sup S and a greatest lower bound m(0,α) = inf S in Rα.Proof. It is a consequence of Theorem 5.Theorem 17. Every continuous function f defined on a compact subset

X of Rα is bounded and reaches its maximum and minimum values.Proof. By Theorem 16, the image S = f(X) is a compact Hausdorff

space of Rα and hence, by Theorem 15, bounded. We apply Theorem 17and obtain M(−α,α) = sup X and m(−α,α) = inf X which by compactnessof S belong to f(S). Since f : X → S is a surjection f , evidently reachesits maximum and minimum values.

78 JU. T. LISICA

Remark 22. If α = ωξ, where ωξ is the initial ordinal number, ξ ≥ 1,then our Rωξ

differs from the space denoted by Rξ in [29] Hausdorff, [1908],because for the latter dim Rξ = 0 and for the former dim Rωξ

= 1.Theorem 18. The class QΩ is an ℵκ-universal linear ordering for all

cardinals ℵκ, κ ∈ On.The Proof is a consequence of Mendelson’s theorem: Qκ is an ℵκ-universal

linear ordering (see [39] and [53], p. 169). �We omit another way to construct a hierarchy of generalized real num-

bers of different powers, which are one-dimensional and continuous but dif-ferent from these presented above, which we had originally planned to showin the present paper. Moreover, algebraic operations would be completelydefined, as in Conway’s approach [20], but through fundamentally differentmethod. The field strucrure will be on that generalized real numbers andeven the completions of fields but all of then except our real numbers R

will be zero-dimensional. There is also a conjecture that Conway’s numbers,which come from Game Theory, are subclasses of these new generalized realnumbers and all Conway’s numbers which is a Field, i.e., with a domain asa class will have a completion which is a FIELD in a more general sense,like a hyperclass. Since this paper is already too long, and a constructionof generalized real numbers with complete operetions is beyond the scopeof skand theory, we plan to publish the new material in another paper inthe near future.

AcknowledgementThe author wishes to thank Professor Philip T. Grier (USA) for help in

editing the manuscript.

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82 JU. T. LISICA

SKAND TEORIJA I NEJZINI PRIMENI.(NOV POGLED MNO�ESTVATA XTO NE SE DOBRO

OSNOVANI)

J. T. Lisica

R e z i m e

Voveden e nov matematiqki objekt nareqen skand, koj vo opxt sluqaj emno�estvo xto ne e dobro osnovano. Skandovite so koneqna dol�ina sese standardni dobro osnovani mno�estva a skandovite so mnogu golemadol�ina (kako hiper-skandot od site ordinali) se hiper-klasi.

Razgledani si i sebe-sliqnite skandovi i tie ja razjasnuvaat reflek-sivnosta na mno�estva, t.e. znaqeǌeto na relacijata X ∈ X ; posebnosebe-slicnite skandovi razgledani kako mno�estva xto ne se dobro osno-vani se sekogax refleksivni, no ne va�i obratno. Postoeǌeto na sebe-sliqni skandovi doka�uva deka dobro poznatite paradoksi od teori-jata na mno�estva voopxto ne se paradoksi i spored toa ne mora da sefatalni za sekoja teorija na mno�estva. Znaqi doka�ana e nekonzis-tentnostga na “mno�estvoto” na Rasel R = {X | X �∈ X} ne so pomox naRaseloviot paradoks (kako xto e tradicionalno daden, i e pogrexen), noso ednostaven metod na maksimalnost (univerzalnost) na R xto vrakana Kantoroviot i e primenet i na drugite paradoksi od teorijata namno�estva.

Definirani se i voopxtenite skandovi i e demonstriran nov pogledna voopxtenata skand-klasa od site ordinali. Specijalno, definirane posledniot ordinal nareqen eschaton.

Slednata primena na teorinata na skandovi e vo opisot na siteepsilon-broevi vo smisla na Kantor. Druga primena e obopxtenatateorija na ednodimenzionalen kontinuum od proizvolen stepen i kon-strukcija na obopxteni realni broevi kako nearhimedova prava od pro-izvolen stepen, i voveduvaǌeto na apsoluten kontinuum i na apsolutnaprava kako hiper-klasa najbliska do klasata od mno�estva.

Mathematical Analysis and Function Theory Department, Russian Univer-

sity of Peoples’ Friendship, Miklukho-Maklay str. 6, 117198 Moscow, Russia,

E-mail address : [email protected]

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